Blockwise Repeated Burst Error Correcting Linear Codes B.K. Dass Department of Mathematics University of Delhi Delhi - 110 007, India dassbk@rediffmail.com Surbhi Madan ∗ Department of Mathematics Shivaji College (University of Delhi) New Delhi - 110 027, India surbhimadan@gmail.com Abstract This paper presents a lower and an upper bound on the number of parity check digits required for a linear code that corrects a single sub- block containing errors which are in the form of 2-repeated bursts of length b or less. An illustration of such kind of codes has been provided. Further, the codes that correct m-repeated bursts of length b or less have also been studied. Keywords: Error locating codes, error correction, burst errors, repeated burst errors AMS Subject Classification: : 94B20, 94B65, 94B25. ∗Corresponding Author Ratio Mathematica 20, 2010 97 I Introduction Error detecting codes and Error correcting codes have been the traditional areas of study in the field of coding techniques on error control in digital data trans- mission. Wolf and Elspas [12] introduced a coding technique, error-locating codes (EL Codes), lying midway between error detection and error correction. In an error locating code, each block of received digits is regarded as being subdivided into mutually exclusive sub-blocks, and codes have been devised that permit the detection of errors occurring within a single sub-block, the sub-block containing errors being identified. In ordinary decision feedback systems using error detec- tion the receiver tests each block of received digits for the presence of errors. If errors are detected, the receiver requests the retransmission of the corrupted block of digits alone and this process is repeated for each incoming block. One drawback of the conventional system is that long block lengths (which are de- sirable for increased coding efficiency) can result in a low data rate when the reception of large amount of data is called for. However, the use of EL codes can soften this conflict between short and long block lengths by providing an additional design parameter. The overall constraint block length can be long to provide efficient coding while the length of the sub-blocks can be relatively short in order to keep the data rate up. Codes developed at the early stages were meant mainly to detect and cor- rect random errors. However, it was observed later that in many channels the likelihood of the occurrence of errors is more in adjacent positions rather than their occurrence in a random manner. In this spirit, Abramson[1] developed codes correcting single and double adjacent errors. The concept of clustered er- rors, commonly called burst errors, was generalized further in the work due to Fire [7]. A burst, also known as an open loop burst, of length b may be defined as follows: Definition 1. A burst of length b is a vector whose all non-zero components are among some b consecutive components, the first and the last of which is non-zero. It was observed that in very busy communication channels, errors repeat themselves. Similar is a situation when errors occur in the form of a burst. The development of codes for such kind of repeated burst errors is useful for Ratio Mathematica 20, 2010 98 improving upon the efficiency of some communication channels. Not only do repeated bursts emerge as a natural generalization of bursts, but considering a recent study by Srinivas, Jain, Saurav and Sikdar [11], where the changes in the neuronal network properties during epileptiform activity in vitro in planar two- dimensional neuronal networks cultured on a multielectrode array using the in vitro model of stroke-induced epilepsy have been explored, we observe that the study of these codes is significant. The study of codes that detect repeated open-loop bursts was initiated by Berardi, Dass and Verma [2] and for correction of such errors by Dass and Verma [6] . An m-repeated burst (open-loop) of length b is defined as follows: Definition 2. An m-repeated burst of length b is a vector of length n whose only non-zero components are confined to m distinct sets of b consecutive components, the first and the last component of each set being non-zero. For example, (001032000020310000313200) is a 3-repeated burst of length 4 over GF(4). In particular, a 2-repeated burst (open-loop) of length b is defined as: Definition 3. A 2-repeated burst of length b is a vector of length n whose only non-zero components are confined to two distinct sets of b consecutive compo- nents, the first and the last component of each set being non-zero. Wolf and Elspas [12] obtained results in the form of bounds over the number of parity-check digits required for binary codes capable of detecting and locating a single sub-block containing random errors. A study of such error locating codes in which errors occur in the form of bursts was made by Dass [3]. Further, these results were extended to the codes correcting burst errors occurring within a sub- block (refer Dass and Tyagi [5]). In our earlier paper [4] the authors obtained bounds over the number of parity-check digits required for codes detecting 2- repeated and m-repeated bursts of length b or less occurring within a single sub- block, the sub-block containing errors being identified. In this paper we extend our study to the correction of repeated bursts occurring within a sub-block. The development of codes correcting repeated burst errors within a sub-block improves the efficiency of the communication channel as it reduces the number of parity Ratio Mathematica 20, 2010 99 check digits required. The results that follow have been described in terms of the following parameters: the block of n digits, consisting of r check digits, and k = n−r information digits, is subdivided into s mutually exclusive sub-blocks, each sub-block containing t = n/s digits. II Bounds for codes correcting 2-repeated bursts In this section, we obtain bounds on the number of parity check digits of a code capable of correcting 2-repeated bursts of length b or less occurring within a single sub-block. We note that an (n,k) linear EL code over GF(q) capable of detecting and locating a single sub-block containing 2-repeated burst of length b or less must satisfy the following two conditions: (i) The syndrome resulting from the occurrence of any 2-repeated burst of length b or less within any one sub-block must be non-zero. (ii) The syndrome resulting from the occurrence of any 2-repeated burst of length b or less within a single sub-block must be distinct from the syndrome resulting likewise from any 2-repeated burst of length b or less within any other sub-block. Further, an (n,k) linear code over GF(q) capable of correcting an error requires the syndromes of any two vectors to be distinct irrespective of whether they belong to the same sub-block or different sub-blocks. So, in order to correct 2-repeated bursts of length b or less lying within a sub-block the following con- ditions need to be satisfied: (iii) The syndrome resulting from the occurrence of any 2-repeated burst of length b or less within a single sub-block must be distinct from the syndrome resulting from any other 2-repeated burst of length b or less within the same sub-block. (iv) The syndrome resulting from the occurrence of any 2-repeated burst of length b or less within a single sub-block must be distinct from the syndrome resulting likewise from any 2-repeated burst of length b or less within any other sub-block. Remark 1. We observe that condition (ii) is the same as condition (iv). Also, for computational purposes condition (i) is taken care of by condition (iii). From this we infer that correction of errors requires more strict conditions than location of Ratio Mathematica 20, 2010 100 errors. So we need to consider conditions (iii) and (iv) or equivalently conditions (ii) and (iii) for correction of the said type of errors. We first obtain a lower bound over the number of parity check digits re- quired for such a code. Theorem 1. The number of check digits r required for an (n,k) linear code over GF(q), subdivided into s sub-blocks of length t each, that corrects 2-repeated bursts of length b or less lying within a single corrupted sub-block is atleast logq { 1+s [ q2b−2 { q + (q − 1)2 ( t− 2b + 2 2 ) + (q − 1) ( t− 2b + 1 1 )} −1 ]} . (1) Proof. Let V be an (n,k) linear code over GF(q) that corrects 2-repeated burst of length b or less within a single corrupted sub-block. The maximum number of distinct syndromes available using r check digits is qr. The proof proceeds by first counting the number of syndromes that are required to be distinct by the two conditions and then setting this number less than or equal to qr. Since the code is capable of correcting all errors which are 2-repeated bursts of length b or less within any single sub-block, any syndrome produced by a 2- repeated burst of length b or less in a given sub-block must be distinct from any such syndrome likewise resulting from another 2-repeated burst of length b or less in the same sub-block(refer to condition (iii)). Moreover, syndromes produced by 2-repeated bursts of length b or less in different sub-blocks must also be distinct by condition (iv). Thus, the syndromes of vectors which are 2-repeated bursts, whether in the same sub-block or in different sub-blocks, must be distinct. Since there are q2b−2 { q + (q − 1)2 ( t− 2b + 2 2 ) + (q − 1) ( t− 2b + 1 1 )} − 1 2-repeated bursts of length b or less within one sub-block of length t, excluding the vector of all zeros( refer Dass and Verma (2008)) and there are s sub-blocks Ratio Mathematica 20, 2010 101 in all, we must have at least 1 + s [ q2b−2 { q + (q − 1)2 ( t− 2b + 2 2 ) + (q − 1) ( t− 2b + 1 1 )} − 1 ] distinct syndromes, including the all zeros syndrome. Therefore, we must have qr ≥ 1 + s [ q2b−2 { q + (q − 1)2 ( t− 2b + 2 2 ) + (q − 1) ( t− 2b + 1 1 )} − 1 ] i.e. r ≥logq { 1 + s [ q2b−2 { q + (q − 1)2 ( t − 2b + 2 2 ) + (q − 1) ( t − 2b + 1 1 )} − 1 ]} . Remark 2. By taking s = 1 the bound obtained in (1) reduces to logq ( q2b−2 [ q + (q − 1)2 ( t− 2b + 2 2 ) + (q − 1) ( t− 2b + 1 1 )]) which coincides with the result for correction of 2-repeated bursts obtained by Dass and Verma(2008). In the following result, we derive another bound on the number of check digits required for the existence of such a code. The proof is based on the tech- nique used to establish Varshamov-Gilbert-Sacks bound by constructing a parity check matrix for such a code ( refer Sacks (1958) also Theorem 4.7, Peterson and Weldon (1972)). This technique not only ensures the existence of such a code but also gives a method for the construction of the code. Theorem 2. An (n,k) linear code over GF(q) capable of correcting 2-repeated burst of length b or less occurring within a single sub-block of length t (4b < t) can always be constructed using r check digits, where r is the smallest integer Ratio Mathematica 20, 2010 102 satisfying the inequality qr > q2(b−1) { q2(b−1) { (q − 1)3 ( t − 4b + 3 3 ) + (q − 1)2 ( t − 4b + 2 2 ) + q(q − 1) ( t − 4b + 1 1 ) + q2 } + { (s − 1) [ (t − 2b + 1)(q − 1) + 1 ] × [ q2(b−1) { q + (q − 1)2 ( t − 2b + 2 2 ) + (q − 1) ( t − 2b + 1 1 )} − 1 ]}} . (2) Proof. We shall prove the result by constructing an appropriate (n − k) × n parity check matrix H for the desired code. Suppose that the columns of the first s− 1 sub-blocks of H and the first j − 1 columns h1,h2, · · · ,hj−1 of the sth sub-block have been appropriately added. We now lay down conditions to add the jth column hj to the s th sub-block as follows: Since the code is to correct 2-repeated bursts of length b or less within a single sub-block, therefore, by condition (iii), the syndrome of any 2-repeated burst in any sub-block must be different from the syndrome resulting from any other such burst within the same sub-block. Therefore the jth column hj can be added provided that hj is not a linear combination of the immediately preceding b − 1 or fewer columns hj−b+1, · · · ,hj−1 of the sth sub-block together with any three distinct sets of b or fewer consecutive columns each from amongst the first j − b columns h1,h2, · · · ,hj−b. In other words, hj 6= (α1hj−b+1 + α2hj−b+2 + · · · + αb−1hj−1)+ 3∑ l=1 (βl1hl1 + βl2hl2 + · · · + βlbhlb ), (3) where αi,βli ∈ GF(q) and lb ≤ j − b. The number of ways in which the coefficients αi can be selected is clearly qb−1. To enumerate the coefficients βi is equivalent to enumerate the number of 3-repeated bursts of length b or less in a vector of length j − b which is (refer Dass and Verma(2008)) q3(b−1) { (q − 1)3 ( j − 4b + 3 3 ) + (q − 1)2 ( j − 4b + 2 2 ) + q(q − 1) ( j − 4b + 1 1 ) + q2 } . Ratio Mathematica 20, 2010 103 Therefore, the total number of possible choices for αi and βi on the R.H.S of (3) is q4(b−1) { (q − 1)3 ( j − 4b + 3 3 ) + (q − 1)2 ( j − 4b + 2 2 ) + q(q − 1) ( j − 4b + 1 1 ) + q2 } . (4) Further, by condition (iv), hj can be added to the s th sub-block provided hj is not a linear combination of the immediately preceding b − 1 or fewer columns together with one set of b or fewer columns from amongst the first j−b columns together with linear combination of any two sets of b or less consecutive columns within any other sub-block. i.e. hj 6= (α1hj−b+1 + α2hj−b+2 + · · · + αb−1hj−1)+ (β1hi + β2hi+1 + · · · + βbhi+b−1)+ (γ1hi1 + γ2hi1+1 + · · · + γbhi1+b−1)+ (δ1hi2 + δ2hi2+1 + · · · + δbhi2+b−1) (5) where αp,βp,γp,δp ∈ GF(q), i + b − 1 ≤ j − b and not all γp and δp are zero. (The last two terms in the above sum correspond to any two sets of b or less consecutive columns within any one of the other sub-block.) The number of ways in which the coefficients αp can be selected is clearly qb−1. To enumerate the coefficients βp is equivalent to enumerate the number of bursts of length b or less in a vector of length j−b which is qb−1[(j−2b+1)(q−1)+1] (refer Fire [7]). Therefore, the total number of possible choices for αp and βp on the R.H.S of (5) is q2(b−1)[(j − 2b + 1)(q − 1) + 1]. (6) Also, the number of linear combinations corresponding to the last two terms on the R.H.S. of (5) is the same as the number of 2-repeated bursts of length b or less within a sub-block of length t, excluding the vector of all zeros; which is ( refer Dass and Verma (2008)) q2b−2 { q + (q − 1)2 ( t− 2b + 2 2 ) + (q − 1) ( t− 2b + 1 1 )} − 1. Since there are s − 1 previously chosen sub-blocks, the number of such linear combinations becomes Ratio Mathematica 20, 2010 104 (s − 1) [ q2b−2 { q + (q − 1)2 ( t − 2b + 2 2 ) + (q − 1) ( t − 2b + 1 1 )} − 1 ] . (7) Thus, the number of linear combinations to which hj can not be equal to is the product computed in expr. (6) and expr. (7). i.e. expr.(6) ×expr.(7). (8) Thus, the total number of linear combinations that hj can not be equal to is the sum of linear combinations in (4) and (8). At worst, all these combinations might yield a distinct sum. Therefore, hj can be added to the sth sub- block of H provided that qr > q2(b−1) { q2(b−1) { (q − 1)3 ( j − 4b + 3 3 ) + (q − 1)2 ( j − 4b + 2 2 ) + q(q − 1) ( j − 4b + 1 1 ) + q2 } + { (s − 1) [ (j − 2b + 1)(q − 1) + 1 ] × [ q2(b−1) { q + (q − 1)2 ( t − 2b + 2 2 ) + (q − 1) ( t − 2b + 1 1 )} − 1 ]}} . For completing the sth sub-block of length t, replacing j by t gives the result as stated in (2). Remark 3. By taking s = 1 in (2) the bound reduces to qr > q4(b−1) { (q − 1)3 ( t − 4b + 3 3 ) + (q − 1)2 ( t − 4b + 2 2 ) + q(q − 1) ( t − 4b + 1 1 ) + q2 } which coincides with the condition for existence of a code correcting 2-repeated bursts of length b or less( refer Dass and Verma(2008)). We conclude this section with an example. Example 1 Consider a (26, 10) binary code with a 16 × 26 parity-check matrix Ratio Mathematica 20, 2010 105 H given by H =   1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 1 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 1 0 1 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 1 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 1 0 1 1 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 1 0 1 0 0 1 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 1 0 0 1 1 0 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 1 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 1 1 0 0 1 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 1 0 0 1 0 0 0   This matrix has been constructed by the synthesis procedure outlined in the proof of Theorem 2 by taking b = 3, s = 2, t = 13 over GF(2) ( MS Excel Program was used for the construction of the matrix). It can be seen from the Table 1 that the syndromes of all distinct 2-repeated bursts of length 3 or less whether in the same sub-block or in different sub-blocks are different, showing thereby that the code that is the null space of this matrix corrects all 2-repeated bursts of length 3 or less occurring within a sub-block. Ratio Mathematica 20, 2010 106 Table 1 Error Patterns - Syndrome vectors Sub-block 1 S.No. Error Vector Syndrome S. no. Error vector Syndrome 1 1111110000000 0000000000000 1111110000000000 44 0111101000000 0000000000000 0111101000000000 2 1110111000000 0000000000000 1110111000000000 45 0111010100000 0000000000000 0111010100000000 3 1110011100000 0000000000000 1110011100000000 46 0111001010000 0000000000000 0111001010000000 4 1110001110000 0000000000000 1110001110000000 47 0111000101000 0000000000000 0111000101000000 5 1110000111000 0000000000000 1110000111000000 48 0111000010100 0000000000000 0111000010100000 6 1110000011100 0000000000000 1110000011100000 49 0111000001010 0000000000000 0111000001010000 7 1110000001110 0000000000000 1110000001110000 50 0111000000101 0000000000000 0111000000101000 8 1110000000111 0000000000000 1110000000111000 51 0111110000000 0000000000000 0111110000000000 9 1111010000000 0000000000000 1111010000000000 52 0111011000000 0000000000000 0111011000000000 10 1110101000000 0000000000000 1110101000000000 53 0111001100000 0000000000000 0111001100000000 11 1110010100000 0000000000000 1110010100000000 54 0111000110000 0000000000000 0111000110000000 12 1110001010000 0000000000000 1110001010000000 55 0111000011000 0000000000000 0111000011000000 13 1110000101000 0000000000000 1110000101000000 56 0111000001100 0000000000000 0111000001100000 14 1110000010100 0000000000000 1110000010100000 57 0111000000110 0000000000000 0111000000110000 15 1110000001010 0000000000000 1110000001010000 58 0111000000011 0000000000000 0111000000011000 16 1110000000101 0000000000000 1110000000101000 59 0111100000000 0000000000000 0111100000000000 17 1111100000000 0000000000000 1111100000000000 60 0111010000000 0000000000000 0111010000000000 18 1110110000000 0000000000000 1110110000000000 61 0111001000000 0000000000000 0111001000000000 19 1110011000000 0000000000000 1110011000000000 62 0111000100000 0000000000000 0111000100000000 20 1110001100000 0000000000000 1110001100000000 63 0111000010000 0000000000000 0111000010000000 21 1110000110000 0000000000000 1110000110000000 64 0111000001000 0000000000000 0111000001000000 22 1110000011000 0000000000000 1110000011000000 65 0111000000100 0000000000000 0111000000100000 23 1110000001100 0000000000000 1110000001100000 66 0111000000010 0000000000000 0111000000010000 24 1110000000110 0000000000000 1110000000110000 67 0111000000001 0000000000000 0111000000001000 25 1110000000011 0000000000000 1110000000011000 68 0111000000000 0000000000000 0111000000000000 26 1111000000000 0000000000000 1111000000000000 69 0011111100000 0000000000000 0011111100000000 27 1110100000000 0000000000000 1110100000000000 70 0011101110000 0000000000000 0011101110000000 28 1110010000000 0000000000000 1110010000000000 71 0011100111000 0000000000000 0011100111000000 29 1110001000000 0000000000000 1110001000000000 72 0011100011100 0000000000000 0011100011100000 30 1110000100000 0000000000000 1110000100000000 73 0011100001110 0000000000000 0011100001110000 31 1110000010000 0000000000000 1110000010000000 74 0011100000111 0000000000000 0011100000111000 32 1110000001000 0000000000000 1110000001000000 75 0011110100000 0000000000000 0011110100000000 33 1110000000100 0000000000000 1110000000100000 76 0011101010000 0000000000000 0011101010000000 34 1110000000010 0000000000000 1110000000010000 77 0011100101000 0000000000000 0011100101000000 35 1110000000001 0000000000000 1110000000001000 78 0011100010100 0000000000000 0011100010100000 36 1110000000000 0000000000000 1110000000000000 79 0011100001010 0000000000000 0011100001010000 37 0111111000000 0000000000000 0111111000000000 80 0011100000101 0000000000000 0011100000101000 38 0111011100000 0000000000000 0111011100000000 81 0011111000000 0000000000000 0011111000000000 39 0111001110000 0000000000000 0111001110000000 82 0011101100000 0000000000000 0011101100000000 40 0111000111000 0000000000000 0111000111000000 83 0011100110000 0000000000000 0011100110000000 41 0111000011100 0000000000000 0111000011100000 84 0011100011000 0000000000000 0011100011000000 42 0111000001110 0000000000000 0111000001110000 85 0011100001100 0000000000000 0011100001100000 43 0111000000111 0000000000000 0111000000111000 86 0011100000110 0000000000000 0011100000110000 Ratio Mathematica 20, 2010 107 Sub-block 1 S.No. Error Vector Syndrome S. no. Error vector Syndrome 87 0011100000011 0000000000000 0011100000011000 134 0000111100000 0000000000000 0000111100000000 88 0011110000000 0000000000000 0011110000000000 135 0000111010000 0000000000000 0000111010000000 89 0011101000000 0000000000000 0011101000000000 136 0000111001000 0000000000000 0000111001000000 90 0011100100000 0000000000000 0011100100000000 137 0000111000100 0000000000000 0000111000100000 91 0011100010000 0000000000000 0011100010000000 138 0000111000010 0000000000000 0000111000010000 92 0011100001000 0000000000000 0011100001000000 139 0000111000001 0000000000000 0000111000001000 93 0011100000100 0000000000000 0011100000100000 140 0000111000000 0000000000000 0000111000000000 94 0011100000010 0000000000000 0011100000010000 141 0000011111100 0000000000000 0000011111100000 95 0011100000001 0000000000000 0011100000001000 142 0000011101110 0000000000000 0000011101110000 96 0011100000000 0000000000000 0011100000000000 143 0000011100111 0000000000000 0000011100111000 97 0001111110000 0000000000000 0001111110000000 144 0000011110100 0000000000000 0000011110100000 98 0001110111000 0000000000000 0001110111000000 145 0000011101010 0000000000000 0000011101010000 99 0001110011100 0000000000000 0001110011100000 146 0000011100101 0000000000000 0000011100101000 100 0001110001110 0000000000000 0001110001110000 147 0000011111000 0000000000000 0000011111000000 101 0001110000111 0000000000000 0001110000111000 148 0000011101100 0000000000000 0000011101100000 102 0001111010000 0000000000000 0001111010000000 149 0000011100110 0000000000000 0000011100110000 103 0001110101000 0000000000000 0001110101000000 150 0000011100011 0000000000000 0000011100011000 104 0001110010100 0000000000000 0001110010100000 151 0000011110000 0000000000000 0000011110000000 105 0001110001010 0000000000000 0001110001010000 152 0000011101000 0000000000000 0000011101000000 106 0001110000101 0000000000000 0001110000101000 153 0000011100100 0000000000000 0000011100100000 107 0001111100000 0000000000000 0001111100000000 154 0000011100010 0000000000000 0000011100010000 108 0001110110000 0000000000000 0001110110000000 155 0000011100001 0000000000000 0000011100001000 109 0001110011000 0000000000000 0001110011000000 156 0000011100000 0000000000000 0000011100000000 110 0001110001100 0000000000000 0001110001100000 157 0000001111110 0000000000000 0000001111110000 111 0001110000110 0000000000000 0001110000110000 158 0000001110111 0000000000000 0000001110111000 112 0001110000011 0000000000000 0001110000011000 159 0000001111010 0000000000000 0000001111010000 113 0001111000000 0000000000000 0001111000000000 160 0000001110101 0000000000000 0000001110101000 114 0001110100000 0000000000000 0001110100000000 161 0000001111100 0000000000000 0000001111100000 115 0001110010000 0000000000000 0001110010000000 162 0000001110110 0000000000000 0000001110110000 116 0001110001000 0000000000000 0001110001000000 163 0000001110011 0000000000000 0000001110011000 117 0001110000100 0000000000000 0001110000100000 164 0000001111000 0000000000000 0000001111000000 118 0001110000010 0000000000000 0001110000010000 165 0000001110100 0000000000000 0000001110100000 119 0001110000001 0000000000000 0001110000001000 166 0000001110010 0000000000000 0000001110010000 120 0001110000000 0000000000000 0001110000000000 167 0000001110001 0000000000000 0000001110001000 121 0000111111000 0000000000000 0000111111000000 168 0000001110000 0000000000000 0000001110000000 122 0000111011100 0000000000000 0000111011100000 169 0000000111111 0000000000000 0000000111111000 123 0000111001110 0000000000000 0000111001110000 170 0000000111101 0000000000000 0000000111101000 124 0000111000111 0000000000000 0000111000111000 171 0000000111110 0000000000000 0000000111110000 125 0000111101000 0000000000000 0000111101000000 172 0000000111011 0000000000000 0000000111011000 126 0000111010100 0000000000000 0000111010100000 173 0000000111100 0000000000000 0000000111100000 127 0000111001010 0000000000000 0000111001010000 174 0000000111010 0000000000000 0000000111010000 128 0000111000101 0000000000000 0000111000101000 175 0000000111001 0000000000000 0000000111001000 129 0000111110000 0000000000000 0000111110000000 176 0000000111000 0000000000000 0000000111000000 130 0000111011000 0000000000000 0000111011000000 177 0000000011111 0000000000000 0000000011111000 131 0000111001100 0000000000000 0000111001100000 178 0000000011101 0000000000000 0000000011101000 132 0000111000110 0000000000000 0000111000110000 179 0000000011110 0000000000000 0000000011110000 133 0000111000011 0000000000000 0000111000011000 180 0000000011100 0000000000000 0000000011100000 Ratio Mathematica 20, 2010 108 Sub-block 1 S.No. Error Vector Syndrome S. no. 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Error vector Syndrome 275 0010100001000 0000000000000 0010100001000000 322 0000101000001 0000000000000 0000101000001000 276 0010100000100 0000000000000 0010100000100000 323 0000101000000 0000000000000 0000101000000000 277 0010100000010 0000000000000 0010100000010000 324 0000010111100 0000000000000 0000010111100000 278 0010100000001 0000000000000 0010100000001000 325 0000010101110 0000000000000 0000010101110000 279 0010100000000 0000000000000 0010100000000000 326 0000010100111 0000000000000 0000010100111000 280 0001011110000 0000000000000 0001011110000000 327 0000010110100 0000000000000 0000010110100000 281 0001010111000 0000000000000 0001010111000000 328 0000010101010 0000000000000 0000010101010000 282 0001010011100 0000000000000 0001010011100000 329 0000010100101 0000000000000 0000010100101000 283 0001010001110 0000000000000 0001010001110000 330 0000010111000 0000000000000 0000010111000000 284 0001010000111 0000000000000 0001010000111000 331 0000010101100 0000000000000 0000010101100000 285 0001011010000 0000000000000 0001011010000000 332 0000010100110 0000000000000 0000010100110000 286 0001010101000 0000000000000 0001010101000000 333 0000010100011 0000000000000 0000010100011000 287 0001010010100 0000000000000 0001010010100000 334 0000010110000 0000000000000 0000010110000000 288 0001010001010 0000000000000 0001010001010000 335 0000010101000 0000000000000 0000010101000000 289 0001010000101 0000000000000 0001010000101000 336 0000010100100 0000000000000 0000010100100000 290 0001011100000 0000000000000 0001011100000000 337 0000010100010 0000000000000 0000010100010000 291 0001010110000 0000000000000 0001010110000000 338 0000010100001 0000000000000 0000010100001000 292 0001010011000 0000000000000 0001010011000000 339 0000010100000 0000000000000 0000010100000000 293 0001010001100 0000000000000 0001010001100000 340 0000001011110 0000000000000 0000001011110000 294 0001010000110 0000000000000 0001010000110000 341 0000001010111 0000000000000 0000001010111000 295 0001010000011 0000000000000 0001010000011000 342 0000001011010 0000000000000 0000001011010000 296 0001011000000 0000000000000 0001011000000000 343 0000001010101 0000000000000 0000001010101000 297 0001010100000 0000000000000 0001010100000000 344 0000001011100 0000000000000 0000001011100000 298 0001010010000 0000000000000 0001010010000000 345 0000001010110 0000000000000 0000001010110000 299 0001010001000 0000000000000 0001010001000000 346 0000001010011 0000000000000 0000001010011000 300 0001010000100 0000000000000 0001010000100000 347 0000001011000 0000000000000 0000001011000000 301 0001010000010 0000000000000 0001010000010000 348 0000001010100 0000000000000 0000001010100000 302 0001010000001 0000000000000 0001010000001000 349 0000001010010 0000000000000 0000001010010000 303 0001010000000 0000000000000 0001010000000000 350 0000001010001 0000000000000 0000001010001000 304 0000101111000 0000000000000 0000101111000000 351 0000001010000 0000000000000 0000001010000000 305 0000101011100 0000000000000 0000101011100000 352 0000000101111 0000000000000 0000000101111000 306 0000101001110 0000000000000 0000101001110000 353 0000000101101 0000000000000 0000000101101000 307 0000101000111 0000000000000 0000101000111000 354 0000000101110 0000000000000 0000000101110000 308 0000101101000 0000000000000 0000101101000000 355 0000000101011 0000000000000 0000000101011000 309 0000101010100 0000000000000 0000101010100000 356 0000000101100 0000000000000 0000000101100000 310 0000101001010 0000000000000 0000101001010000 357 0000000101010 0000000000000 0000000101010000 311 0000101000101 0000000000000 0000101000101000 358 0000000101001 0000000000000 0000000101001000 312 0000101110000 0000000000000 0000101110000000 359 0000000101000 0000000000000 0000000101000000 313 0000101011000 0000000000000 0000101011000000 360 0000000010111 0000000000000 0000000010111000 314 0000101001100 0000000000000 0000101001100000 361 0000000010101 0000000000000 0000000010101000 315 0000101000110 0000000000000 0000101000110000 362 0000000010110 0000000000000 0000000010110000 316 0000101000011 0000000000000 0000101000011000 363 0000000010100 0000000000000 0000000010100000 317 0000101100000 0000000000000 0000101100000000 364 0000000001011 0000000000000 0000000001011000 318 0000101010000 0000000000000 0000101010000000 365 0000000001010 0000000000000 0000000001010000 319 0000101001000 0000000000000 0000101001000000 366 0000000000101 0000000000000 0000000000101000 320 0000101000100 0000000000000 0000101000100000 367 1101110000000 0000000000000 1101110000000000 321 0000101000010 0000000000000 0000101000010000 368 1100111000000 0000000000000 1100111000000000 Ratio Mathematica 20, 2010 110 Sub-block 1 S.No. Error Vector Syndrome S. no. 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Error vector Syndrome 463 0001101110000 0000000000000 0001101110000000 510 0000011010100 0000000000000 0000011010100000 464 0001100111000 0000000000000 0001100111000000 511 0000011001010 0000000000000 0000011001010000 465 0001100011100 0000000000000 0001100011100000 512 0000011000101 0000000000000 0000011000101000 466 0001100001110 0000000000000 0001100001110000 513 0000011011000 0000000000000 0000011011000000 467 0001100000111 0000000000000 0001100000111000 514 0000011001100 0000000000000 0000011001100000 468 0001101010000 0000000000000 0001101010000000 515 0000011000110 0000000000000 0000011000110000 469 0001100101000 0000000000000 0001100101000000 516 0000011000011 0000000000000 0000011000011000 470 0001100010100 0000000000000 0001100010100000 517 0000011010000 0000000000000 0000011010000000 471 0001100001010 0000000000000 0001100001010000 518 0000011001000 0000000000000 0000011001000000 472 0001100000101 0000000000000 0001100000101000 519 0000011000100 0000000000000 0000011000100000 473 0001101100000 0000000000000 0001101100000000 520 0000011000010 0000000000000 0000011000010000 474 0001100110000 0000000000000 0001100110000000 521 0000011000001 0000000000000 0000011000001000 475 0001100011000 0000000000000 0001100011000000 522 0000011000000 0000000000000 0000011000000000 476 0001100001100 0000000000000 0001100001100000 523 0000001101110 0000000000000 0000001101110000 477 0001100000110 0000000000000 0001100000110000 524 0000001100111 0000000000000 0000001100111000 478 0001100000011 0000000000000 0001100000011000 525 0000001101010 0000000000000 0000001101010000 479 0001101000000 0000000000000 0001101000000000 526 0000001100101 0000000000000 0000001100101000 480 0001100100000 0000000000000 0001100100000000 527 0000001101100 0000000000000 0000001101100000 481 0001100010000 0000000000000 0001100010000000 528 0000001100110 0000000000000 0000001100110000 482 0001100001000 0000000000000 0001100001000000 529 0000001100011 0000000000000 0000001100011000 483 0001100000100 0000000000000 0001100000100000 530 0000001101000 0000000000000 0000001101000000 484 0001100000010 0000000000000 0001100000010000 531 0000001100100 0000000000000 0000001100100000 485 0001100000001 0000000000000 0001100000001000 532 0000001100010 0000000000000 0000001100010000 486 0001100000000 0000000000000 0001100000000000 533 0000001100001 0000000000000 0000001100001000 487 0000110111000 0000000000000 0000110111000000 534 0000001100000 0000000000000 0000001100000000 488 0000110011100 0000000000000 0000110011100000 535 0000000110111 0000000000000 0000000110111000 489 0000110001110 0000000000000 0000110001110000 536 0000000110101 0000000000000 0000000110101000 490 0000110000111 0000000000000 0000110000111000 537 0000000110110 0000000000000 0000000110110000 491 0000110101000 0000000000000 0000110101000000 538 0000000110011 0000000000000 0000000110011000 492 0000110010100 0000000000000 0000110010100000 539 0000000110100 0000000000000 0000000110100000 493 0000110001010 0000000000000 0000110001010000 540 0000000110010 0000000000000 0000000110010000 494 0000110000101 0000000000000 0000110000101000 541 0000000110001 0000000000000 0000000110001000 495 0000110110000 0000000000000 0000110110000000 542 0000000110000 0000000000000 0000000110000000 496 0000110011000 0000000000000 0000110011000000 543 0000000011011 0000000000000 0000000011011000 497 0000110001100 0000000000000 0000110001100000 544 0000000011001 0000000000000 0000000011001000 498 0000110000110 0000000000000 0000110000110000 545 0000000011010 0000000000000 0000000011010000 499 0000110000011 0000000000000 0000110000011000 546 0000000011000 0000000000000 0000000011000000 500 0000110100000 0000000000000 0000110100000000 547 0000000001101 0000000000000 0000000001101000 501 0000110010000 0000000000000 0000110010000000 548 0000000001100 0000000000000 0000000001100000 502 0000110001000 0000000000000 0000110001000000 549 0000000000110 0000000000000 0000000000110000 503 0000110000100 0000000000000 0000110000100000 550 0000000000011 0000000000000 0000000000011000 504 0000110000010 0000000000000 0000110000010000 551 1001110000000 0000000000000 1001110000000000 505 0000110000001 0000000000000 0000110000001000 552 1000111000000 0000000000000 1000111000000000 506 0000110000000 0000000000000 0000110000000000 553 1000011100000 0000000000000 1000011100000000 507 0000011011100 0000000000000 0000011011100000 554 1000001110000 0000000000000 1000001110000000 508 0000011001110 0000000000000 0000011001110000 555 1000000111000 0000000000000 1000000111000000 509 0000011000111 0000000000000 0000011000111000 556 1000000011100 0000000000000 1000000011100000 Ratio Mathematica 20, 2010 112 Sub-block 1 S.No. Error Vector Syndrome S. no. Error vector Syndrome 557 1000000001110 0000000000000 1000000001110000 604 0100000110000 0000000000000 0100000110000000 558 1000000000111 0000000000000 1000000000111000 605 0100000011000 0000000000000 0100000011000000 559 1001010000000 0000000000000 1001010000000000 606 0100000001100 0000000000000 0100000001100000 560 1000101000000 0000000000000 1000101000000000 607 0100000000110 0000000000000 0100000000110000 561 1000010100000 0000000000000 1000010100000000 608 0100000000011 0000000000000 0100000000011000 562 1000001010000 0000000000000 1000001010000000 609 0100100000000 0000000000000 0100100000000000 563 1000000101000 0000000000000 1000000101000000 610 0100010000000 0000000000000 0100010000000000 564 1000000010100 0000000000000 1000000010100000 611 0100001000000 0000000000000 0100001000000000 565 1000000001010 0000000000000 1000000001010000 612 0100000100000 0000000000000 0100000100000000 566 1000000000101 0000000000000 1000000000101000 613 0100000010000 0000000000000 0100000010000000 567 1001100000000 0000000000000 1001100000000000 614 0100000001000 0000000000000 0100000001000000 568 1000110000000 0000000000000 1000110000000000 615 0100000000100 0000000000000 0100000000100000 569 1000011000000 0000000000000 1000011000000000 616 0100000000010 0000000000000 0100000000010000 570 1000001100000 0000000000000 1000001100000000 617 0100000000001 0000000000000 0100000000001000 571 1000000110000 0000000000000 1000000110000000 618 0100000000000 0000000000000 0100000000000000 572 1000000011000 0000000000000 1000000011000000 619 0010011100000 0000000000000 0010011100000000 573 1000000001100 0000000000000 1000000001100000 620 0010001110000 0000000000000 0010001110000000 574 1000000000110 0000000000000 1000000000110000 621 0010000111000 0000000000000 0010000111000000 575 1000000000011 0000000000000 1000000000011000 622 0010000011100 0000000000000 0010000011100000 576 1001000000000 0000000000000 1001000000000000 623 0010000001110 0000000000000 0010000001110000 577 1000100000000 0000000000000 1000100000000000 624 0010000000111 0000000000000 0010000000111000 578 1000010000000 0000000000000 1000010000000000 625 0010010100000 0000000000000 0010010100000000 579 1000001000000 0000000000000 1000001000000000 626 0010001010000 0000000000000 0010001010000000 580 1000000100000 0000000000000 1000000100000000 627 0010000101000 0000000000000 0010000101000000 581 1000000010000 0000000000000 1000000010000000 628 0010000010100 0000000000000 0010000010100000 582 1000000001000 0000000000000 1000000001000000 629 0010000001010 0000000000000 0010000001010000 583 1000000000100 0000000000000 1000000000100000 630 0010000000101 0000000000000 0010000000101000 584 1000000000010 0000000000000 1000000000010000 631 0010011000000 0000000000000 0010011000000000 585 1000000000001 0000000000000 1000000000001000 632 0010001100000 0000000000000 0010001100000000 586 1000000000000 0000000000000 1000000000000000 633 0010000110000 0000000000000 0010000110000000 587 0100111000000 0000000000000 0100111000000000 634 0010000011000 0000000000000 0010000011000000 588 0100011100000 0000000000000 0100011100000000 635 0010000001100 0000000000000 0010000001100000 589 0100001110000 0000000000000 0100001110000000 636 0010000000110 0000000000000 0010000000110000 590 0100000111000 0000000000000 0100000111000000 637 0010000000011 0000000000000 0010000000011000 591 0100000011100 0000000000000 0100000011100000 638 0010010000000 0000000000000 0010010000000000 592 0100000001110 0000000000000 0100000001110000 639 0010001000000 0000000000000 0010001000000000 593 0100000000111 0000000000000 0100000000111000 640 0010000100000 0000000000000 0010000100000000 594 0100101000000 0000000000000 0100101000000000 641 0010000010000 0000000000000 0010000010000000 595 0100010100000 0000000000000 0100010100000000 642 0010000001000 0000000000000 0010000001000000 596 0100001010000 0000000000000 0100001010000000 643 0010000000100 0000000000000 0010000000100000 597 0100000101000 0000000000000 0100000101000000 644 0010000000010 0000000000000 0010000000010000 598 0100000010100 0000000000000 0100000010100000 645 0010000000001 0000000000000 0010000000001000 599 0100000001010 0000000000000 0100000001010000 646 0010000000000 0000000000000 0010000000000000 600 0100000000101 0000000000000 0100000000101000 647 0001001110000 0000000000000 0001001110000000 601 0100110000000 0000000000000 0100110000000000 648 0001000111000 0000000000000 0001000111000000 602 0100011000000 0000000000000 0100011000000000 649 0001000011100 0000000000000 0001000011100000 603 0100001100000 0000000000000 0100001100000000 650 0001000001110 0000000000000 0001000001110000 Ratio Mathematica 20, 2010 113 Sub-block 1 S.No. Error Vector Syndrome S. no. Error vector Syndrome 651 0001000000111 0000000000000 0001000000111000 694 0000010010100 0000000000000 0000010010100000 652 0001001010000 0000000000000 0001001010000000 695 0000010001010 0000000000000 0000010001010000 653 0001000101000 0000000000000 0001000101000000 696 0000010000101 0000000000000 0000010000101000 654 0001000010100 0000000000000 0001000010100000 697 0000010011000 0000000000000 0000010011000000 655 0001000001010 0000000000000 0001000001010000 698 0000010001100 0000000000000 0000010001100000 656 0001000000101 0000000000000 0001000000101000 699 0000010000110 0000000000000 0000010000110000 657 0001001100000 0000000000000 0001001100000000 700 0000010000011 0000000000000 0000010000011000 658 0001000110000 0000000000000 0001000110000000 701 0000010010000 0000000000000 0000010010000000 659 0001000011000 0000000000000 0001000011000000 702 0000010001000 0000000000000 0000010001000000 660 0001000001100 0000000000000 0001000001100000 703 0000010000100 0000000000000 0000010000100000 661 0001000000110 0000000000000 0001000000110000 704 0000010000010 0000000000000 0000010000010000 662 0001000000011 0000000000000 0001000000011000 705 0000010000001 0000000000000 0000010000001000 663 0001001000000 0000000000000 0001001000000000 706 0000010000000 0000000000000 0000010000000000 664 0001000100000 0000000000000 0001000100000000 707 0000001001110 0000000000000 0000001001110000 665 0001000010000 0000000000000 0001000010000000 708 0000001000111 0000000000000 0000001000111000 666 0001000001000 0000000000000 0001000001000000 709 0000001001010 0000000000000 0000001001010000 667 0001000000100 0000000000000 0001000000100000 710 0000001000101 0000000000000 0000001000101000 668 0001000000010 0000000000000 0001000000010000 711 0000001001100 0000000000000 0000001001100000 669 0001000000001 0000000000000 0001000000001000 712 0000001000110 0000000000000 0000001000110000 670 0001000000000 0000000000000 0001000000000000 713 0000001000011 0000000000000 0000001000011000 671 0000100111000 0000000000000 0000100111000000 714 0000001001000 0000000000000 0000001001000000 672 0000100011100 0000000000000 0000100011100000 715 0000001000100 0000000000000 0000001000100000 673 0000100001110 0000000000000 0000100001110000 716 0000001000010 0000000000000 0000001000010000 674 0000100000111 0000000000000 0000100000111000 717 0000001000001 0000000000000 0000001000001000 675 0000100101000 0000000000000 0000100101000000 718 0000001000000 0000000000000 0000001000000000 676 0000100010100 0000000000000 0000100010100000 719 0000000100111 0000000000000 0000000100111000 677 0000100001010 0000000000000 0000100001010000 720 0000000100101 0000000000000 0000000100101000 678 0000100000101 0000000000000 0000100000101000 721 0000000100110 0000000000000 0000000100110000 679 0000100110000 0000000000000 0000100110000000 722 0000000100011 0000000000000 0000000100011000 680 0000100011000 0000000000000 0000100011000000 723 0000000100100 0000000000000 0000000100100000 681 0000100001100 0000000000000 0000100001100000 724 0000000100010 0000000000000 0000000100010000 682 0000100000110 0000000000000 0000100000110000 725 0000000100001 0000000000000 0000000100001000 683 0000100000011 0000000000000 0000100000011000 726 0000000100000 0000000000000 0000000100000000 684 0000100100000 0000000000000 0000100100000000 727 0000000010011 0000000000000 0000000010011000 685 0000100010000 0000000000000 0000100010000000 728 0000000010001 0000000000000 0000000010001000 686 0000100001000 0000000000000 0000100001000000 729 0000000010010 0000000000000 0000000010010000 687 0000100000100 0000000000000 0000100000100000 730 0000000010000 0000000000000 0000000010000000 688 0000100000010 0000000000000 0000100000010000 731 0000000001001 0000000000000 0000000001001000 689 0000100000001 0000000000000 0000100000001000 732 0000000001000 0000000000000 0000000001000000 690 0000100000000 0000000000000 0000100000000000 733 0000000000100 0000000000000 0000000000100000 691 0000010011100 0000000000000 0000010011100000 734 0000000000010 0000000000000 0000000000010000 692 0000010001110 0000000000000 0000010001110000 735 0000000000001 0000000000000 0000000000001000 693 0000010000111 0000000000000 0000010000111000 Ratio Mathematica 20, 2010 114 Sub-block 2 S.No. Error Vector Syndrome S. no. Error vector Syndrome 736 0000000000000 1111110000000 0000111111111101 779 0000000000000 0111101000000 0000010010000100 737 0000000000000 1110111000000 0000111111101010 780 0000000000000 0111010100000 0000111111100101 738 0000000000000 1110011100000 0000111111110110 781 0000000000000 0111001010000 0000100101110100 739 0000000000000 1110001110000 0000111110110100 782 0000000000000 0111000101000 0001011010011100 740 0000000000000 1110000111000 0001110110110000 783 0000000000000 0111000010100 0000000000110111 741 0000000000000 1110000011100 0001001000100000 784 0000000000000 0111000001010 0011001010011010 742 0000000000000 1110000001110 0011101111001010 785 0000000000000 0111000000101 0001100000001101 743 0000000000000 1110000000111 0011101011001011 786 0000000000000 0111110000000 0000111111111001 744 0000000000000 1111010000000 0000101101101101 787 0000000000000 0111011000000 0000100100110110 745 0000000000000 1110101000000 0000011011001000 788 0000000000000 0111001100000 0000010010011000 746 0000000000000 1110010100000 0000110110101001 789 0000000000000 0111000110000 0000111110100111 747 0000000000000 1110001010000 0000101100111000 790 0000000000000 0111000011000 0001101101110000 748 0000000000000 1110000101000 0001010011010000 791 0000000000000 0111000001100 0001100100001100 749 0000000000000 1110000010100 0000001001111011 792 0000000000000 0111000000110 0010100111011101 750 0000000000000 1110000001010 0011000011010110 793 0000000000000 0111000000011 0011001110011011 751 0000000000000 1110000000101 0001101001000001 794 0000000000000 0111100000000 0000011011011011 752 0000000000000 1111100000000 0000011011011111 795 0000000000000 0111010000000 0000101101101001 753 0000000000000 1110110000000 0000110110110101 796 0000000000000 0111001000000 0000000000010100 754 0000000000000 1110011000000 0000101101111010 797 0000000000000 0111000100000 0000011011000111 755 0000000000000 1110001100000 0000011011010100 798 0000000000000 0111000010000 0000101100101011 756 0000000000000 1110000110000 0000110111101011 799 0000000000000 0111000001000 0001001000010000 757 0000000000000 1110000011000 0001100100111100 800 0000000000000 0111000000100 0000100101010111 758 0000000000000 1110000001100 0001101101000000 801 0000000000000 0111000000010 0010001011000001 759 0000000000000 1110000000110 0010101110010001 802 0000000000000 0111000000001 0001001100010001 760 0000000000000 1110000000011 0011000111010111 803 0000000000000 0111000000000 0000001001001011 761 0000000000000 1111000000000 0000001001001111 804 0000000000000 0011111100000 0000100100101000 762 0000000000000 1110100000000 0000010010010111 805 0000000000000 0011101110000 0000100101101010 763 0000000000000 1110010000000 0000100100100101 806 0000000000000 0011100111000 0001101101101110 764 0000000000000 1110001000000 0000001001011000 807 0000000000000 0011100011100 0001010011111110 765 0000000000000 1110000100000 0000010010001011 808 0000000000000 0011100001110 0011110100010100 766 0000000000000 1110000010000 0000100101100111 809 0000000000000 0011100000111 0011110000010101 767 0000000000000 1110000001000 0001000001011100 810 0000000000000 0011110100000 0000101101110111 768 0000000000000 1110000000100 0000101100011011 811 0000000000000 0011101010000 0000110111100110 769 0000000000000 1110000000010 0010000010001101 812 0000000000000 0011100101000 0001001000001110 770 0000000000000 1110000000001 0001000101011101 813 0000000000000 0011100010100 0000010010100101 771 0000000000000 1110000000000 0000000000000111 814 0000000000000 0011100001010 0011011000001000 772 0000000000000 0111111000000 0000110110100110 815 0000000000000 0011100000101 0001110010011111 773 0000000000000 0111011100000 0000110110111010 816 0000000000000 0011111000000 0000110110100100 774 0000000000000 0111001110000 0000110111111000 817 0000000000000 0011101100000 0000000000001010 775 0000000000000 0111000111000 0001111111111100 818 0000000000000 0011100110000 0000101100110101 776 0000000000000 0111000011100 0001000001101100 819 0000000000000 0011100011000 0001111111100010 777 0000000000000 0111000001110 0011100110000110 820 0000000000000 0011100001100 0001110110011110 778 0000000000000 0111000000111 0011100010000111 821 0000000000000 0011100000110 0010110101001111 Ratio Mathematica 20, 2010 115 Sub-block 2 S.No. Error Vector Syndrome S. no. Error vector Syndrome 822 0000000000000 0011100000011 0011011100001001 869 0000000000000 0000111100000 0000101101100001 823 0000000000000 0011110000000 0000111111111011 870 0000000000000 0000111010000 0000011010001101 824 0000000000000 0011101000000 0000010010000110 871 0000000000000 0000111001000 0001111110110110 825 0000000000000 0011100100000 0000001001010101 872 0000000000000 0000111000100 0000010011110001 826 0000000000000 0011100010000 0000111110111001 873 0000000000000 0000111000010 0010111101100111 827 0000000000000 0011100001000 0001011010000010 874 0000000000000 0000111000001 0001111010110111 828 0000000000000 0011100000100 0000110111000101 875 0000000000000 0000111000000 0000111111101101 829 0000000000000 0011100000010 0010011001010011 876 0000000000000 0000011111100 0001110111010110 830 0000000000000 0011100000001 0001011110000011 877 0000000000000 0000011101110 0011010000111100 831 0000000000000 0011100000000 0000011011011001 878 0000000000000 0000011100111 0011010100111101 832 0000000000000 0001111110000 0000000001001001 879 0000000000000 0000011110100 0000110110001101 833 0000000000000 0001110111000 0001001001001101 880 0000000000000 0000011101010 0011111100100000 834 0000000000000 0001110011100 0001110111011101 881 0000000000000 0000011100101 0001010110110111 835 0000000000000 0001110001110 0011010000110111 882 0000000000000 0000011111000 0001011011001010 836 0000000000000 0001110000111 0011010100110110 883 0000000000000 0000011101100 0001010010110110 837 0000000000000 0001111010000 0000010011000101 884 0000000000000 0000011100110 0010010001100111 838 0000000000000 0001110101000 0001101100101101 885 0000000000000 0000011100011 0011111000100001 839 0000000000000 0001110010100 0000110110000110 886 0000000000000 0000011110000 0000011010010001 840 0000000000000 0001110001010 0011111100101011 887 0000000000000 0000011101000 0001111110101010 841 0000000000000 0001110000101 0001010110111100 888 0000000000000 0000011100100 0000010011101101 842 0000000000000 0001111100000 0000100100101001 889 0000000000000 0000011100010 0010111101111011 843 0000000000000 0001110110000 0000001000010110 890 0000000000000 0000011100001 0001111010101011 844 0000000000000 0001110011000 0001011011000001 891 0000000000000 0000011100000 0000111111110001 845 0000000000000 0001110001100 0001010010111101 892 0000000000000 0000001111110 0011010001111110 846 0000000000000 0001110000110 0010010001101100 893 0000000000000 0000001110111 0011010101111111 847 0000000000000 0001110000011 0011111000101010 894 0000000000000 0000001111010 0011111101100010 848 0000000000000 0001111000000 0000110110100101 895 0000000000000 0000001110101 0001010111110101 849 0000000000000 0001110100000 0000101101110110 896 0000000000000 0000001111100 0001010011110100 850 0000000000000 0001110010000 0000011010011010 897 0000000000000 0000001110110 0010010000100101 851 0000000000000 0001110001000 0001111110100001 898 0000000000000 0000001110011 0011111001100011 852 0000000000000 0001110000100 0000010011100110 899 0000000000000 0000001111000 0001111111101000 853 0000000000000 0001110000010 0010111101110000 900 0000000000000 0000001110100 0000010010101111 854 0000000000000 0001110000001 0001111010100000 901 0000000000000 0000001110010 0010111100111001 855 0000000000000 0001110000000 0000111111111010 902 0000000000000 0000001110001 0001111011101001 856 0000000000000 0000111111000 0001001001011010 903 0000000000000 0000001110000 0000111110110011 857 0000000000000 0000111011100 0001110111001010 904 0000000000000 0000000111111 0010011101111011 858 0000000000000 0000111001110 0011010000100000 905 0000000000000 0000000111101 0000011111110001 859 0000000000000 0000111000111 0011010100100001 906 0000000000000 0000000111110 0011011000100001 860 0000000000000 0000111101000 0001101100111010 907 0000000000000 0000000111011 0010110001100111 861 0000000000000 0000111010100 0000110110010001 908 0000000000000 0000000111100 0001011010101011 862 0000000000000 0000111001010 0011111100111100 909 0000000000000 0000000111010 0011110100111101 863 0000000000000 0000111000101 0001010110101011 910 0000000000000 0000000111001 0000110011101101 864 0000000000000 0000111110000 0000001000000001 911 0000000000000 0000000111000 0001110110110111 865 0000000000000 0000111011000 0001011011010110 912 0000000000000 0000000011111 0010001111110111 866 0000000000000 0000111001100 0001010010101010 913 0000000000000 0000000011101 0000001101111101 867 0000000000000 0000111000110 0010010001111011 914 0000000000000 0000000011110 0011001010101101 868 0000000000000 0000111000011 0011111000111101 915 0000000000000 0000000011100 0001001000100111 Ratio Mathematica 20, 2010 116 Sub-block 2 S.No. Error Vector Syndrome S. no. Error vector Syndrome 916 0000000000000 0000000001111 0010101010010111 963 0000000000000 0101010100000 0000111111100100 917 0000000000000 0000000001110 0011101111001101 964 0000000000000 0101001010000 0000100101110101 918 0000000000000 0000000000111 0011101011001100 965 0000000000000 0101000101000 0001011010011101 919 0000000000000 1011110000000 0000111111111111 966 0000000000000 0101000010100 0000000000110110 920 0000000000000 1010111000000 0000111111101000 967 0000000000000 0101000001010 0011001010011011 921 0000000000000 1010011100000 0000111111110100 968 0000000000000 0101000000101 0001100000001100 922 0000000000000 1010001110000 0000111110110110 969 0000000000000 0101110000000 0000111111111000 923 0000000000000 1010000111000 0001110110110010 970 0000000000000 0101011000000 0000100100110111 924 0000000000000 1010000011100 0001001000100010 971 0000000000000 0101001100000 0000010010011001 925 0000000000000 1010000001110 0011101111001000 972 0000000000000 0101000110000 0000111110100110 926 0000000000000 1010000000111 0011101011001001 973 0000000000000 0101000011000 0001101101110001 927 0000000000000 1011010000000 0000101101101111 974 0000000000000 0101000001100 0001100100001101 928 0000000000000 1010101000000 0000011011001010 975 0000000000000 0101000000110 0010100111011100 929 0000000000000 1010010100000 0000110110101011 976 0000000000000 0101000000011 0011001110011010 930 0000000000000 1010001010000 0000101100111010 977 0000000000000 0101100000000 0000011011011010 931 0000000000000 1010000101000 0001010011010010 978 0000000000000 0101010000000 0000101101101000 932 0000000000000 1010000010100 0000001001111001 979 0000000000000 0101001000000 0000000000010101 933 0000000000000 1010000001010 0011000011010100 980 0000000000000 0101000100000 0000011011000110 934 0000000000000 1010000000101 0001101001000011 981 0000000000000 0101000010000 0000101100101010 935 0000000000000 1011100000000 0000011011011101 982 0000000000000 0101000001000 0001001000010001 936 0000000000000 1010110000000 0000110110110111 983 0000000000000 0101000000100 0000100101010110 937 0000000000000 1010011000000 0000101101111000 984 0000000000000 0101000000010 0010001011000000 938 0000000000000 1010001100000 0000011011010110 985 0000000000000 0101000000001 0001001100010000 939 0000000000000 1010000110000 0000110111101001 986 0000000000000 0101000000000 0000001001001010 940 0000000000000 1010000011000 0001100100111110 987 0000000000000 0010111100000 0000101101100000 941 0000000000000 1010000001100 0001101101000010 988 0000000000000 0010101110000 0000101100100010 942 0000000000000 1010000000110 0010101110010011 989 0000000000000 0010100111000 0001100100100110 943 0000000000000 1010000000011 0011000111010101 990 0000000000000 0010100011100 0001011010110110 944 0000000000000 1011000000000 0000001001001101 991 0000000000000 0010100001110 0011111101011100 945 0000000000000 1010100000000 0000010010010101 992 0000000000000 0010100000111 0011111001011101 946 0000000000000 1010010000000 0000100100100111 993 0000000000000 0010110100000 0000100100111111 947 0000000000000 1010001000000 0000001001011010 994 0000000000000 0010101010000 0000111110101110 948 0000000000000 1010000100000 0000010010001001 995 0000000000000 0010100101000 0001000001000110 949 0000000000000 1010000010000 0000100101100101 996 0000000000000 0010100010100 0000011011101101 950 0000000000000 1010000001000 0001000001011110 997 0000000000000 0010100001010 0011010001000000 951 0000000000000 1010000000100 0000101100011001 998 0000000000000 0010100000101 0001111011010111 952 0000000000000 1010000000010 0010000010001111 999 0000000000000 0010111000000 0000111111101100 953 0000000000000 1010000000001 0001000101011111 1000 0000000000000 0010101100000 0000001001000010 954 0000000000000 1010000000000 0000000000000101 1001 0000000000000 0010100110000 0000100101111101 955 0000000000000 0101111000000 0000110110100111 1002 0000000000000 0010100011000 0001110110101010 956 0000000000000 0101011100000 0000110110111011 1003 0000000000000 0010100001100 0001111111010110 957 0000000000000 0101001110000 0000110111111001 1004 0000000000000 0010100000110 0010111100000111 958 0000000000000 0101000111000 0001111111111101 1005 0000000000000 0010100000011 0011010101000001 959 0000000000000 0101000011100 0001000001101101 1006 0000000000000 0010110000000 0000110110110011 960 0000000000000 0101000001110 0011100110000111 1007 0000000000000 0010101000000 0000011011001110 961 0000000000000 0101000000111 0011100010000110 1008 0000000000000 0010100100000 0000000000011101 962 0000000000000 0101101000000 0000010010000101 1009 0000000000000 0010100010000 0000110111110001 Ratio Mathematica 20, 2010 117 Sub-block 2 S.No. Error Vector Syndrome S. no. Error vector Syndrome 1010 0000000000000 0010100001000 0001010011001010 1057 0000000000000 0000101000001 0001011110010101 1011 0000000000000 0010100000100 0000111110001101 1058 0000000000000 0000101000000 0000011011001111 1012 0000000000000 0010100000010 0010010000011011 1059 0000000000000 0000010111100 0001111110001001 1013 0000000000000 0010100000001 0001010111001011 1060 0000000000000 0000010101110 0011011001100011 1014 0000000000000 0010100000000 0000010010010001 1061 0000000000000 0000010100111 0011011101100010 1015 0000000000000 0001011110000 0000010011011001 1062 0000000000000 0000010110100 0000111111010010 1016 0000000000000 0001010111000 0001011011011101 1063 0000000000000 0000010101010 0011110101111111 1017 0000000000000 0001010011100 0001100101001101 1064 0000000000000 0000010100101 0001011111101000 1018 0000000000000 0001010001110 0011000010100111 1065 0000000000000 0000010111000 0001010010010101 1019 0000000000000 0001010000111 0011000110100110 1066 0000000000000 0000010101100 0001011011101001 1020 0000000000000 0001011010000 0000000001010101 1067 0000000000000 0000010100110 0010011000111000 1021 0000000000000 0001010101000 0001111110111101 1068 0000000000000 0000010100011 0011110001111110 1022 0000000000000 0001010010100 0000100100010110 1069 0000000000000 0000010110000 0000010011001110 1023 0000000000000 0001010001010 0011101110111011 1070 0000000000000 0000010101000 0001110111110101 1024 0000000000000 0001010000101 0001000100101100 1071 0000000000000 0000010100100 0000011010110010 1025 0000000000000 0001011100000 0000110110111001 1072 0000000000000 0000010100010 0010110100100100 1026 0000000000000 0001010110000 0000011010000110 1073 0000000000000 0000010100001 0001110011110100 1027 0000000000000 0001010011000 0001001001010001 1074 0000000000000 0000010100000 0000110110101110 1028 0000000000000 0001010001100 0001000000101101 1075 0000000000000 0000001011110 0011000011110010 1029 0000000000000 0001010000110 0010000011111100 1076 0000000000000 0000001010111 0011000111110011 1030 0000000000000 0001010000011 0011101010111010 1077 0000000000000 0000001011010 0011101111101110 1031 0000000000000 0001011000000 0000100100110101 1078 0000000000000 0000001010101 0001000101111001 1032 0000000000000 0001010100000 0000111111100110 1079 0000000000000 0000001011100 0001000001111000 1033 0000000000000 0001010010000 0000001000001010 1080 0000000000000 0000001010110 0010000010101001 1034 0000000000000 0001010001000 0001101100110001 1081 0000000000000 0000001010011 0011101011101111 1035 0000000000000 0001010000100 0000000001110110 1082 0000000000000 0000001011000 0001101101100100 1036 0000000000000 0001010000010 0010101111100000 1083 0000000000000 0000001010100 0000000000100011 1037 0000000000000 0001010000001 0001101000110000 1084 0000000000000 0000001010010 0010101110110101 1038 0000000000000 0001010000000 0000101101101010 1085 0000000000000 0000001010001 0001101001100101 1039 0000000000000 0000101111000 0001101101111000 1086 0000000000000 0000001010000 0000101100111111 1040 0000000000000 0000101011100 0001010011101000 1087 0000000000000 0000000101111 0010111000011011 1041 0000000000000 0000101001110 0011110100000010 1088 0000000000000 0000000101101 0000111010010001 1042 0000000000000 0000101000111 0011110000000011 1089 0000000000000 0000000101110 0011111101000001 1043 0000000000000 0000101101000 0001001000011000 1090 0000000000000 0000000101011 0010010100000111 1044 0000000000000 0000101010100 0000010010110011 1091 0000000000000 0000000101100 0001111111001011 1045 0000000000000 0000101001010 0011011000011110 1092 0000000000000 0000000101010 0011010001011101 1046 0000000000000 0000101000101 0001110010001001 1093 0000000000000 0000000101001 0000010110001101 1047 0000000000000 0000101110000 0000101100100011 1094 0000000000000 0000000101000 0001010011010111 1048 0000000000000 0000101011000 0001111111110100 1095 0000000000000 0000000010111 0011001110101100 1049 0000000000000 0000101001100 0001110110001000 1096 0000000000000 0000000010101 0001001100100110 1050 0000000000000 0000101000110 0010110101011001 1097 0000000000000 0000000010110 0010001011110110 1051 0000000000000 0000101000011 0011011100011111 1098 0000000000000 0000000010100 0000001001111100 1052 0000000000000 0000101100000 0000001001000011 1099 0000000000000 0000000001011 0010000110001011 1053 0000000000000 0000101010000 0000111110101111 1100 0000000000000 0000000001010 0011000011010001 1054 0000000000000 0000101001000 0001011010010100 1101 0000000000000 0000000000101 0001101001000110 1055 0000000000000 0000101000100 0000110111010011 1102 0000000000000 1101110000000 0000111111111100 1056 0000000000000 0000101000010 0010011001000101 1103 0000000000000 1100111000000 0000111111101011 Ratio Mathematica 20, 2010 118 Sub-block 2 S.No. Error Vector Syndrome S. no. Error vector Syndrome 1104 0000000000000 1100011100000 0000111111110111 1151 0000000000000 0110000000101 0001101001000101 1105 0000000000000 1100001110000 0000111110110101 1152 0000000000000 0110110000000 0000110110110001 1106 0000000000000 1100000111000 0001110110110001 1153 0000000000000 0110011000000 0000101101111110 1107 0000000000000 1100000011100 0001001000100001 1154 0000000000000 0110001100000 0000011011010000 1108 0000000000000 1100000001110 0011101111001011 1155 0000000000000 0110000110000 0000110111101111 1109 0000000000000 1100000000111 0011101011001010 1156 0000000000000 0110000011000 0001100100111000 1110 0000000000000 1101010000000 0000101101101100 1157 0000000000000 0110000001100 0001101101000100 1111 0000000000000 1100101000000 0000011011001001 1158 0000000000000 0110000000110 0010101110010101 1112 0000000000000 1100010100000 0000110110101000 1159 0000000000000 0110000000011 0011000111010011 1113 0000000000000 1100001010000 0000101100111001 1160 0000000000000 0110100000000 0000010010010011 1114 0000000000000 1100000101000 0001010011010001 1161 0000000000000 0110010000000 0000100100100001 1115 0000000000000 1100000010100 0000001001111010 1162 0000000000000 0110001000000 0000001001011100 1116 0000000000000 1100000001010 0011000011010111 1163 0000000000000 0110000100000 0000010010001111 1117 0000000000000 1100000000101 0001101001000000 1164 0000000000000 0110000010000 0000100101100011 1118 0000000000000 1101100000000 0000011011011110 1165 0000000000000 0110000001000 0001000001011000 1119 0000000000000 1100110000000 0000110110110100 1166 0000000000000 0110000000100 0000101100011111 1120 0000000000000 1100011000000 0000101101111011 1167 0000000000000 0110000000010 0010000010001001 1121 0000000000000 1100001100000 0000011011010101 1168 0000000000000 0110000000001 0001000101011001 1122 0000000000000 1100000110000 0000110111101010 1169 0000000000000 0110000000000 0000000000000011 1123 0000000000000 1100000011000 0001100100111101 1170 0000000000000 0011011100000 0000110110111000 1124 0000000000000 1100000001100 0001101101000001 1171 0000000000000 0011001110000 0000110111111010 1125 0000000000000 1100000000110 0010101110010000 1172 0000000000000 0011000111000 0001111111111110 1126 0000000000000 1100000000011 0011000111010110 1173 0000000000000 0011000011100 0001000001101110 1127 0000000000000 1101000000000 0000001001001110 1174 0000000000000 0011000001110 0011100110000100 1128 0000000000000 1100100000000 0000010010010110 1175 0000000000000 0011000000111 0011100010000101 1129 0000000000000 1100010000000 0000100100100100 1176 0000000000000 0011010100000 0000111111100111 1130 0000000000000 1100001000000 0000001001011001 1177 0000000000000 0011001010000 0000100101110110 1131 0000000000000 1100000100000 0000010010001010 1178 0000000000000 0011000101000 0001011010011110 1132 0000000000000 1100000010000 0000100101100110 1179 0000000000000 0011000010100 0000000000110101 1133 0000000000000 1100000001000 0001000001011101 1180 0000000000000 0011000001010 0011001010011000 1134 0000000000000 1100000000100 0000101100011010 1181 0000000000000 0011000000101 0001100000001111 1135 0000000000000 1100000000010 0010000010001100 1182 0000000000000 0011011000000 0000100100110100 1136 0000000000000 1100000000001 0001000101011100 1183 0000000000000 0011001100000 0000010010011010 1137 0000000000000 1100000000000 0000000000000110 1184 0000000000000 0011000110000 0000111110100101 1138 0000000000000 0110111000000 0000111111101110 1185 0000000000000 0011000011000 0001101101110010 1139 0000000000000 0110011100000 0000111111110010 1186 0000000000000 0011000001100 0001100100001110 1140 0000000000000 0110001110000 0000111110110000 1187 0000000000000 0011000000110 0010100111011111 1141 0000000000000 0110000111000 0001110110110100 1188 0000000000000 0011000000011 0011001110011001 1142 0000000000000 0110000011100 0001001000100100 1189 0000000000000 0011010000000 0000101101101011 1143 0000000000000 0110000001110 0011101111001110 1190 0000000000000 0011001000000 0000000000010110 1144 0000000000000 0110000000111 0011101011001111 1191 0000000000000 0011000100000 0000011011000101 1145 0000000000000 0110101000000 0000011011001100 1192 0000000000000 0011000010000 0000101100101001 1146 0000000000000 0110010100000 0000110110101101 1193 0000000000000 0011000001000 0001001000010010 1147 0000000000000 0110001010000 0000101100111100 1194 0000000000000 0011000000100 0000100101010101 1148 0000000000000 0110000101000 0001010011010100 1195 0000000000000 0011000000010 0010001011000011 1149 0000000000000 0110000010100 0000001001111111 1196 0000000000000 0011000000001 0001001100010011 1150 0000000000000 0110000001010 0011000011010010 1197 0000000000000 0011000000000 0000001001001001 Ratio Mathematica 20, 2010 119 Sub-block 2 S.No. Error Vector Syndrome S. no. Error vector Syndrome 1198 0000000000000 0001101110000 0000100101101011 1245 0000000000000 0000011010100 0000100100000001 1199 0000000000000 0001100111000 0001101101101111 1246 0000000000000 0000011001010 0011101110101100 1200 0000000000000 0001100011100 0001010011111111 1247 0000000000000 0000011000101 0001000100111011 1201 0000000000000 0001100001110 0011110100010101 1248 0000000000000 0000011011000 0001001001000110 1202 0000000000000 0001100000111 0011110000010100 1249 0000000000000 0000011001100 0001000000111010 1203 0000000000000 0001101010000 0000110111100111 1250 0000000000000 0000011000110 0010000011101011 1204 0000000000000 0001100101000 0001001000001111 1251 0000000000000 0000011000011 0011101010101101 1205 0000000000000 0001100010100 0000010010100100 1252 0000000000000 0000011010000 0000001000011101 1206 0000000000000 0001100001010 0011011000001001 1253 0000000000000 0000011001000 0001101100100110 1207 0000000000000 0001100000101 0001110010011110 1254 0000000000000 0000011000100 0000000001100001 1208 0000000000000 0001101100000 0000000000001011 1255 0000000000000 0000011000010 0010101111110111 1209 0000000000000 0001100110000 0000101100110100 1256 0000000000000 0000011000001 0001101000100111 1210 0000000000000 0001100011000 0001111111100011 1257 0000000000000 0000011000000 0000101101111101 1211 0000000000000 0001100001100 0001110110011111 1258 0000000000000 0000001101110 0011110100011110 1212 0000000000000 0001100000110 0010110101001110 1259 0000000000000 0000001100111 0011110000011111 1213 0000000000000 0001100000011 0011011100001000 1260 0000000000000 0000001101010 0011011000000010 1214 0000000000000 0001101000000 0000010010000111 1261 0000000000000 0000001100101 0001110010010101 1215 0000000000000 0001100100000 0000001001010100 1262 0000000000000 0000001101100 0001110110010100 1216 0000000000000 0001100010000 0000111110111000 1263 0000000000000 0000001100110 0010110101000101 1217 0000000000000 0001100001000 0001011010000011 1264 0000000000000 0000001100011 0011011100000011 1218 0000000000000 0001100000100 0000110111000100 1265 0000000000000 0000001101000 0001011010001000 1219 0000000000000 0001100000010 0010011001010010 1266 0000000000000 0000001100100 0000110111001111 1220 0000000000000 0001100000001 0001011110000010 1267 0000000000000 0000001100010 0010011001011001 1221 0000000000000 0001100000000 0000011011011000 1268 0000000000000 0000001100001 0001011110001001 1222 0000000000000 0000110111000 0001000000000101 1269 0000000000000 0000001100000 0000011011010011 1223 0000000000000 0000110011100 0001111110010101 1270 0000000000000 0000000110111 0011011100100000 1224 0000000000000 0000110001110 0011011001111111 1271 0000000000000 0000000110101 0001011110101010 1225 0000000000000 0000110000111 0011011101111110 1272 0000000000000 0000000110110 0010011001111010 1226 0000000000000 0000110101000 0001100101100101 1273 0000000000000 0000000110011 0011110000111100 1227 0000000000000 0000110010100 0000111111001110 1274 0000000000000 0000000110100 0000011011110000 1228 0000000000000 0000110001010 0011110101100011 1275 0000000000000 0000000110010 0010110101100110 1229 0000000000000 0000110000101 0001011111110100 1276 0000000000000 0000000110001 0001110010110110 1230 0000000000000 0000110110000 0000000001011110 1277 0000000000000 0000000110000 0000110111101100 1231 0000000000000 0000110011000 0001010010001001 1278 0000000000000 0000000011011 0010100011101011 1232 0000000000000 0000110001100 0001011011110101 1279 0000000000000 0000000011001 0000100001100001 1233 0000000000000 0000110000110 0010011000100100 1280 0000000000000 0000000011010 0011100110110001 1234 0000000000000 0000110000011 0011110001100010 1281 0000000000000 0000000011000 0001100100111011 1235 0000000000000 0000110100000 0000100100111110 1282 0000000000000 0000000001101 0000101000011101 1236 0000000000000 0000110010000 0000010011010010 1283 0000000000000 0000000001100 0001101101000111 1237 0000000000000 0000110001000 0001110111101001 1284 0000000000000 0000000000110 0010101110010110 1238 0000000000000 0000110000100 0000011010101110 1285 0000000000000 0000000000011 0011000111010000 1239 0000000000000 0000110000010 0010110100111000 1286 0000000000000 1001110000000 0000111111111110 1240 0000000000000 0000110000001 0001110011101000 1287 0000000000000 1000111000000 0000111111101001 1241 0000000000000 0000110000000 0000110110110010 1288 0000000000000 1000011100000 0000111111110101 1242 0000000000000 0000011011100 0001100101011010 1289 0000000000000 1000001110000 0000111110110111 1243 0000000000000 0000011001110 0011000010110000 1290 0000000000000 1000000111000 0001110110110011 1244 0000000000000 0000011000111 0011000110110001 1291 0000000000000 1000000011100 0001001000100011 Ratio Mathematica 20, 2010 120 Sub-block 2 S.No. Error Vector Syndrome S. no. Error vector Syndrome 1292 0000000000000 1000000001110 0011101111001001 1339 0000000000000 0100000110000 0000110111101110 1293 0000000000000 1000000000111 0011101011001000 1340 0000000000000 0100000011000 0001100100111001 1294 0000000000000 1001010000000 0000101101101110 1341 0000000000000 0100000001100 0001101101000101 1295 0000000000000 1000101000000 0000011011001011 1342 0000000000000 0100000000110 0010101110010100 1296 0000000000000 1000010100000 0000110110101010 1343 0000000000000 0100000000011 0011000111010010 1297 0000000000000 1000001010000 0000101100111011 1344 0000000000000 0100100000000 0000010010010010 1298 0000000000000 1000000101000 0001010011010011 1345 0000000000000 0100010000000 0000100100100000 1299 0000000000000 1000000010100 0000001001111000 1346 0000000000000 0100001000000 0000001001011101 1300 0000000000000 1000000001010 0011000011010101 1347 0000000000000 0100000100000 0000010010001110 1301 0000000000000 1000000000101 0001101001000010 1348 0000000000000 0100000010000 0000100101100010 1302 0000000000000 1001100000000 0000011011011100 1349 0000000000000 0100000001000 0001000001011001 1303 0000000000000 1000110000000 0000110110110110 1350 0000000000000 0100000000100 0000101100011110 1304 0000000000000 1000011000000 0000101101111001 1351 0000000000000 0100000000010 0010000010001000 1305 0000000000000 1000001100000 0000011011010111 1352 0000000000000 0100000000001 0001000101011000 1306 0000000000000 1000000110000 0000110111101000 1353 0000000000000 0100000000000 0000000000000010 1307 0000000000000 1000000011000 0001100100111111 1354 0000000000000 0010011100000 0000111111110000 1308 0000000000000 1000000001100 0001101101000011 1355 0000000000000 0010001110000 0000111110110010 1309 0000000000000 1000000000110 0010101110010010 1356 0000000000000 0010000111000 0001110110110110 1310 0000000000000 1000000000011 0011000111010100 1357 0000000000000 0010000011100 0001001000100110 1311 0000000000000 1001000000000 0000001001001100 1358 0000000000000 0010000001110 0011101111001100 1312 0000000000000 1000100000000 0000010010010100 1359 0000000000000 0010000000111 0011101011001101 1313 0000000000000 1000010000000 0000100100100110 1360 0000000000000 0010010100000 0000110110101111 1314 0000000000000 1000001000000 0000001001011011 1361 0000000000000 0010001010000 0000101100111110 1315 0000000000000 1000000100000 0000010010001000 1362 0000000000000 0010000101000 0001010011010110 1316 0000000000000 1000000010000 0000100101100100 1363 0000000000000 0010000010100 0000001001111101 1317 0000000000000 1000000001000 0001000001011111 1364 0000000000000 0010000001010 0011000011010000 1318 0000000000000 1000000000100 0000101100011000 1365 0000000000000 0010000000101 0001101001000111 1319 0000000000000 1000000000010 0010000010001110 1366 0000000000000 0010011000000 0000101101111100 1320 0000000000000 1000000000001 0001000101011110 1367 0000000000000 0010001100000 0000011011010010 1321 0000000000000 1000000000000 0000000000000100 1368 0000000000000 0010000110000 0000110111101101 1322 0000000000000 0100111000000 0000111111101111 1369 0000000000000 0010000011000 0001100100111010 1323 0000000000000 0100011100000 0000111111110011 1370 0000000000000 0010000001100 0001101101000110 1324 0000000000000 0100001110000 0000111110110001 1371 0000000000000 0010000000110 0010101110010111 1325 0000000000000 0100000111000 0001110110110101 1372 0000000000000 0010000000011 0011000111010001 1326 0000000000000 0100000011100 0001001000100101 1373 0000000000000 0010010000000 0000100100100011 1327 0000000000000 0100000001110 0011101111001111 1374 0000000000000 0010001000000 0000001001011110 1328 0000000000000 0100000000111 0011101011001110 1375 0000000000000 0010000100000 0000010010001101 1329 0000000000000 0100101000000 0000011011001101 1376 0000000000000 0010000010000 0000100101100001 1330 0000000000000 0100010100000 0000110110101100 1377 0000000000000 0010000001000 0001000001011010 1331 0000000000000 0100001010000 0000101100111101 1378 0000000000000 0010000000100 0000101100011101 1332 0000000000000 0100000101000 0001010011010101 1379 0000000000000 0010000000010 0010000010001011 1333 0000000000000 0100000010100 0000001001111110 1380 0000000000000 0010000000001 0001000101011011 1334 0000000000000 0100000001010 0011000011010011 1381 0000000000000 0010000000000 0000000000000001 1335 0000000000000 0100000000101 0001101001000100 1382 0000000000000 0001001110000 0000110111111011 1336 0000000000000 0100110000000 0000110110110000 1383 0000000000000 0001000111000 0001111111111111 1337 0000000000000 0100011000000 0000101101111111 1384 0000000000000 0001000011100 0001000001101111 1338 0000000000000 0100001100000 0000011011010001 1385 0000000000000 0001000001110 0011100110000101 Ratio Mathematica 20, 2010 121 Sub-block 2 S.No. Error Vector Syndrome S. no. Error vector Syndrome 1386 0000000000000 0001000000111 0011100010000100 1429 0000000000000 0000010010100 0000101101011110 1387 0000000000000 0001001010000 0000100101110111 1430 0000000000000 0000010001010 0011100111110011 1388 0000000000000 0001000101000 0001011010011111 1431 0000000000000 0000010000101 0001001101100100 1389 0000000000000 0001000010100 0000000000110100 1432 0000000000000 0000010011000 0001000000011001 1390 0000000000000 0001000001010 0011001010011001 1433 0000000000000 0000010001100 0001001001100101 1391 0000000000000 0001000000101 0001100000001110 1434 0000000000000 0000010000110 0010001010110100 1392 0000000000000 0001001100000 0000010010011011 1435 0000000000000 0000010000011 0011100011110010 1393 0000000000000 0001000110000 0000111110100100 1436 0000000000000 0000010010000 0000000001000010 1394 0000000000000 0001000011000 0001101101110011 1437 0000000000000 0000010001000 0001100101111001 1395 0000000000000 0001000001100 0001100100001111 1438 0000000000000 0000010000100 0000001000111110 1396 0000000000000 0001000000110 0010100111011110 1439 0000000000000 0000010000010 0010100110101000 1397 0000000000000 0001000000011 0011001110011000 1440 0000000000000 0000010000001 0001100001111000 1398 0000000000000 0001001000000 0000000000010111 1441 0000000000000 0000010000000 0000100100100010 1399 0000000000000 0001000100000 0000011011000100 1442 0000000000000 0000001001110 0011100110010010 1400 0000000000000 0001000010000 0000101100101000 1443 0000000000000 0000001000111 0011100010010011 1401 0000000000000 0001000001000 0001001000010011 1444 0000000000000 0000001001010 0011001010001110 1402 0000000000000 0001000000100 0000100101010100 1445 0000000000000 0000001000101 0001100000011001 1403 0000000000000 0001000000010 0010001011000010 1446 0000000000000 0000001001100 0001100100011000 1404 0000000000000 0001000000001 0001001100010010 1447 0000000000000 0000001000110 0010100111001001 1405 0000000000000 0001000000000 0000001001001000 1448 0000000000000 0000001000011 0011001110001111 1406 0000000000000 0000100111000 0001100100100111 1449 0000000000000 0000001001000 0001001000000100 1407 0000000000000 0000100011100 0001011010110111 1450 0000000000000 0000001000100 0000100101000011 1408 0000000000000 0000100001110 0011111101011101 1451 0000000000000 0000001000010 0010001011010101 1409 0000000000000 0000100000111 0011111001011100 1452 0000000000000 0000001000001 0001001100000101 1410 0000000000000 0000100101000 0001000001000111 1453 0000000000000 0000001000000 0000001001011111 1411 0000000000000 0000100010100 0000011011101100 1454 0000000000000 0000000100111 0011111001000000 1412 0000000000000 0000100001010 0011010001000001 1455 0000000000000 0000000100101 0001111011001010 1413 0000000000000 0000100000101 0001111011010110 1456 0000000000000 0000000100110 0010111100011010 1414 0000000000000 0000100110000 0000100101111100 1457 0000000000000 0000000100011 0011010101011100 1415 0000000000000 0000100011000 0001110110101011 1458 0000000000000 0000000100100 0000111110010000 1416 0000000000000 0000100001100 0001111111010111 1459 0000000000000 0000000100010 0010010000000110 1417 0000000000000 0000100000110 0010111100000110 1460 0000000000000 0000000100001 0001010111010110 1418 0000000000000 0000100000011 0011010101000000 1461 0000000000000 0000000100000 0000010010001100 1419 0000000000000 0000100100000 0000000000011100 1462 0000000000000 0000000010011 0011100010110000 1420 0000000000000 0000100010000 0000110111110000 1463 0000000000000 0000000010001 0001100000111010 1421 0000000000000 0000100001000 0001010011001011 1464 0000000000000 0000000010010 0010100111101010 1422 0000000000000 0000100000100 0000111110001100 1465 0000000000000 0000000010000 0000100101100000 1423 0000000000000 0000100000010 0010010000011010 1466 0000000000000 0000000001001 0000000100000001 1424 0000000000000 0000100000001 0001010111001010 1467 0000000000000 0000000001000 0001000001011011 1425 0000000000000 0000100000000 0000010010010000 1468 0000000000000 0000000000100 0000101100011100 1426 0000000000000 0000010011100 0001101100000101 1469 0000000000000 0000000000010 0010000010001010 1427 0000000000000 0000010001110 0011001011101111 1470 0000000000000 0000000000001 0001000101011010 1428 0000000000000 0000010000111 0011001111101110 Ratio Mathematica 20, 2010 122 III Bounds for codes correcting m-repeated bursts In this section, we extend the results of previous section to the case of m-repeated bursts of length b or less occurring within a single sub-block. Similar to the case of correction of 2-repeated burst occurring within a sub-block, an (n,k) linear code over GF(q) capable of correcting any sub-block containing m-repeated burst of length b or less must satisfy the following two conditions: (v) The syndrome resulting from the occurrence of any m-repeated burst of length b or less within a single sub-block must be distinct from the syndrome resulting from any other m-repeated burst within the same sub-block. (vi) The syndrome resulting from the occurrence of any m-repeated burst of length b or less within a single sub-block must be distinct from the syndrome resulting likewise from any m-repeated burst of length b or less within any other sub-block. We now present a lower bound on the number of parity check digits required for such a code. Theorem 3. The number of check digits r required for an (n,k) linear code over GF(q), subdivided into s sub-blocks of length t each, that corrects m-repeated bursts of length b or less lying within a single corrupted sub-block is atleast logq { 1 + s [ qm(b−1) (( t−mb + m m ) (q − 1)m+ m−1∑ l=0 ( t−mb + l l ) (q − 1)lqm−1−l ) − 1 ]} . (9) Proof. The proof of this result is on the similar lines as that of proof of Theorem 1 so we omit the proof. Remark 4. By taking s = 1 the bound obtained in (9) reduces to logq { qm(b−1) (( t−mb + m m ) (q − 1)m + m−1∑ l=0 ( t−mb + l l ) (q − 1)lqm−1−l )} . Ratio Mathematica 20, 2010 123 which coincides with the result for correction of m-repeated burst obtained by Dass and Verma(2008). Remark 5. For m = 2, the bound obtained in (9) coincides with the bound obtained in (1) for the case of 2-repeated bursts. In particular, for m = 1, the bound in (9) reduces to 1 + s ( qb−1 ( (t− b + 1)(q − 1) + 1 ) − 1 ) which reduces to the result for correction of burst of length b or less within a sub-block. In the following result, we present another bound on the number of check digits required for the existence of the code considered in Theorem 3. Theorem 4. An (n,k) linear code over GF(q) capable of correcting m-repeated burst of length b or less occurring within a single sub-block of length t (2mb < t) can always be constructed using r check digits where r is the smallest integer satisfying the inequality qr >qm(b−1) { qm(b−1) ( (q − 1)2m−1 ( t − 2mb + (2m − 1) 2m − 1 ) + 2m−2∑ l=0 (q − 1)lq2m−2−l ( t − 2mb + l l )) + ( (s − 1) × [ (q − 1)m−1 ( t − mb + (m − 1) m − 1 ) + m−2∑ l=0 (q − 1)lqm−2−l ( t − mb + l l )] × [ qm(b−1) (( t − mb + m m ) (q − 1)m+ m−1∑ l=0 ( t − mb + l l ) (q − 1)lqm−1−l ) − 1 ])} . (10) Proof. As in Theorem 3, we omit the proof of this result since it can be derived on lines similar to that of Theorem 2. Ratio Mathematica 20, 2010 124 Remark 6. By taking s = 1 in (10) the bound reduces to qr > q2m(b−1) ( (q − 1)2m−1 ( t − 2mb + (2m − 1) 2m − 1 ) + 2m−2∑ l=0 (q − 1)lq2m−2−l ( t − 2mb + l l )) which coincides with the sufficient condition for existence of a code correcting m-repeated bursts( refer Dass and Verma(2008)). Remark 7. For m = 2, the result obtained in Theorem 4 coincides with the result in Theorem 2, for the case of 2-repeated burst of length b or less. For m = 1, the bound in (10) reduces to qb−1 ( qb−1 [ (q − 1)(t− 2b + 1) + 1 ] + (s− 1) [ qb−1 ( (t− b + 1)(q − 1) + 1 ) − 1 ]) which is the condition for existence of a code correcting bursts of length b or less within a sub-block. References [1] Abramson, N.M., A class of systematic codes for non-independent errors, IRE Trans. on Information Theory IT 5(4) (1959) 150-157. [2] Berardi, L., Dass, B.K. and Verma, Rashmi, On 2-repeated burst error de- tecting codes, Journal of Statistical Theory and Practice 3(2) (2009) 381-391. [3] Dass, B.K., Burst Error Locating Codes, J. Inf. and Optimization Sciences 3(1) (1982) 77-80. [4] Dass, B.K., Madan, Surbhi, Repeated Burst Error Locating Linear Codes, Communicated. [5] Dass, B.K., Verma, Rashmi, Repeated burst error correcting linear codes, Asian-European Journal of Mathematics 1(3) (2008) 303-335. Ratio Mathematica 20, 2010 125 [6] Fire, P., A class of multiple-error-correcting binary codes for non- independent errors, Sylvania Report RSL-E-2, Sylvania Reconnaissance Sys- tems Laboratory, Mountain View, Calif (1959). [7] Hamming, R.W., Error-detecting and error-correcting codes. Bell System Technical Journal 29 (1950) 147- 160. [8] Peterson, W.W., Weldon, E.J., Jr., Error-Correcting Codes, 2nd ed., The MIT Press, Mass (1972). [9] Sacks, G.E., Multiple error correction by means of parity-checks, IRE Trans. Inform. Theory IT 4 (1958) 145-147. [10] Srinivas, K.V., Jain, R., Saurav, S. and Sikdar, S.K., Small-world network topology of hippocampal neuronal network is lost, in an in vitro glutamate injury model of epilepsy, European Journal of Neuroscience, 25 (2007) 3276- 3286. [11] Wolf, J., Elspas B., Error-locating codes–A new concept in error control, IEEE Transactions on Information Theory 9(2) (1963) 113-117. Ratio Mathematica 20, 2010 126