ISSN 2148-838Xhttp://dx.doi.org/10.13069/jacodesmath.327360 J. Algebra Comb. Discrete Appl. 4(3) • 227–234 Received: 14 May 2016 Accepted: 27 September 2016 Journal of Algebra Combinatorics Discrete Structures and Applications Some new ternary linear codes∗ Research Article Rumen Daskalov, Plamen Hristov Abstract: Let an [n, k, d]q code be a linear code of length n, dimension k and minimum Hamming distance d over GF(q). One of the most important problems in coding theory is to construct codes with optimal minimum distances. In this paper 22 new ternary linear codes are presented. Two of them are optimal. All new codes improve the respective lower bounds in [11]. 2010 MSC: 94B05, 94B65 Keywords: Ternary linear codes, Construction X 1. Introduction Let an [n,k,d]q code be a linear code of length n, dimension k and minimum Hamming distance d over a finite field GF(q). One of the most important and fundamental problems in coding theory is to find the optimal values of the parameters of a linear code. This optimization problem can be formulated in a couple of ways. For example, for fixed q,n and k we may wish to maximize the minimum distance d; or for given q,k and d to minimize the block length n. Let dq(n,k) denote the largest value of d for which there exists an [n,k,d]q code, and nq(k,d) be the smallest value of n for which there exists an [n,k,d]q code. Then an [nq(k,d),k,d]q code is called length- optimal and an [n,k,dq(n,k)]q code is called distance-optimal. Both length-optimal and distance-optimal codes are called optimal codes. The problem of finding the parameters of optimal codes is a very difficult one and has two aspects - one involves the construction of new codes with better minimum distances and the other is proving the nonexistence of codes with given parameters. It has been solved only over small finite fields for small dimensions and co-dimensions. Computer search is often used in looking for codes with better minimum distances, but it is a well known fact that computing the minimum distance of a linear code is an NP-hard problem [15]. Since it is ∗ This work was partially supported by the Bulgarian Ministry of Education and Science under Contract in TU– Gabrovo. Rumen Daskalov (Corresponding Author), Plamen Hristov; Department of Mathematics, Technical University of Gabrovo, Bulgaria (email: daskalov@tugab.bg, plhristov9@gmail.com). 227 http://orcid.org/0000-0001-7441-4757 http://orcid.org/0000-0002-7350-4061 R. Daskalov, P. Hristov / J. Algebra Comb. Discrete Appl. 4(3) (2017) 227–234 not possible to carry out exhaustive searches for linear codes with large dimension, it is natural to focus one’s effort on subclasses of linear codes, having rich mathematical structures. Quasi-cyclic (QC) codes are known to have such a structure and it has been shown in recent years that this subclass contains many new good linear codes ([1, 4–10, 12–14] and [E. Metodieva, N. Daskalova, Generating generalized necklaces and new quasi-cyclic codes, in preparation, 2017]). Grassl [11] maintains a table with lower and upper bounds on minimum distances of linear codes over small finite fields GF(q) (q ≤ 9). When the constructed code has a minimum distance equal to the upper bound, it is optimal and there is no place for improvement in the table. When there is a gap between the minimum distance of the best-known code and the upper bound on the minimum distance, this is indicated in the table by listing both values - dl and du. Many of the best-known codes in these tables are QC codes. A code that attains a lower bound in the table is called a good code. A code that improves a lower bound in the table will be called a new code. Another online table of linear codes is also maintained by Chen. Chen’s table [3] contains only good and best-known quasi-cyclic and quasi-twisted codes (q ≤ 13). These two databases are updated when new codes are discovered. The remainder of the paper is organized as follows. In Section 2, some basic definitions and facts on QC codes are presented. In Section 3, sixteen good one-generator QC codes (p ≥ 2) are constructed using an algebraic-combinatorial computer search. In Section 4 (Theorem 4.1), we use the codes presented in section 3, along with construction X, to obtain seventeen new ternary linear codes. In Theorem 4.2 five new codes are also presented. 2. Quasi-cyclic codes A code C is said to be quasi-cyclic (QC or p-QC) if a cyclic shift of a codeword by p positions results in another codeword. A cyclic shift of an m-tuple (x0,x1, . . . ,xm−1) is the m-tuple (xm−1,x0, . . . ,xm−2). The blocklength n of a p-QC code is a multiple of p, so that n = pm. A matrix B of the form B =   b0 b1 b2 · · · bm−2 bm−1 bm−1 b0 b1 · · · bm−3 bm−2 bm−2 bm−1 b0 · · · bm−4 bm−3 ... ... ... ... ... b1 b2 b3 · · · bm−1 b0   , (1) is called a circulant matrix. A class of QC codes can be constructed from m×m circulant matrices. In this case, the generator matrix G can be represented as G = [B1, B2, ... , Bp] , (2) where Bi is a circulant matrix. The algebra of m×m circulant matrices over GF(q) is isomorphic to the algebra of polynomials in the ring GF(q)[x]/(xm−1), with B being mapped to the polynomial, b(x) = b0+b1x+b2x2+· · ·+bm−1xm−1, formed from the entries in the first row of B. The bi(x)’s associated with a QC code are called the defining polynomials. If the defining polynomials bi(x) contain a common factor which is also a factor of xm −1, then the QC code is called degenerate. The dimension k of the QC code is equal to the degree of h(x), where h(x) = xm −1 gcd{xm −1,b0(x),b1(x), · · · ,bp−1(x)} . (3) 228 http://orcid.org/0000-0001-7441-4757 http://orcid.org/0000-0002-7350-4061 R. Daskalov, P. Hristov / J. Algebra Comb. Discrete Appl. 4(3) (2017) 227–234 If the polynomial h(x) has degree m, the dimension of the code is m, and (2) is a generator matrix. If deg(h(x)) = k < m, a generator matrix for the code can be constructed by deleting m − k rows of (2). Let the defining polynomials of the code C have the following form d1(x) = g(x), d2(x) = g(x)f2(x), · · · , dp(x) = g(x)fp(x), (4) where g(x)|(xm − 1),g(x),fi(x) ∈ GF(q)[x]/(xm − 1), (fi(x),(xm − 1)/g(x)) = 1 and deg fi(x) < m−deg g(x) for all 1 ≤ i ≤ p. Then C is a degenerate, one-generator QC code having n = mp, and k = m−deg g(x) (see [14]). In our constructions we will use the following well-known theorems. Theorem 2.1 (Construction X). [2] Given an [n,k,d]q code C1, and an [n,k− l,d + s]q subcode C2, we can construct an [n+a,k,d+s]q code C when we have an [a,l,s]q code C3 (by appending codewords from the latter code to cosets of the second code in the first code). Theorem 2.2 (Construction XX). [2] Let an [n,k,d]q code C have two subcodes C1 and C2 of dimensions k−k1 and k−k2 and append tails from a [ai,ki,δi]q code to the codewords of C , where the two tails of codewords correspond to the coset of Ci(i = 1,2) it is in. If C1, C2 and C1 ⋂ C2 have minimum distance d1, d2 and d0, respectively, then there exists an [n+a1 +a2,k,min(d0,d1 +δ2,d2 +δ1,d+δ1 +δ2)]q code. 3. Good QC codes In this section sixteen good one-generator QC codes (p ≤ 4) are constructed using a non-exhaustive algebraic-combinatorial computer search, similar to that in [1, 4–6, 8–10, 14]. An important feature of these codes is that they have good subcodes and can be used for construction X. We have restricted our search to one-generator QC codes with defining polynomials of the form (4). Example 3.1. : Let m = 35. The factorization of the polynomial x35 −1 over GF(3) is x35 −1 = 5∏ i=1 pi(x), where p1(x) = x 12 + x10 + 2x8 + x7 + x5 + 2x4 + x3 + 2x2 + 2x + 1 p2(x) = x 12 + 2x11 + 2x10 + x9 + 2x8 + x7 + x5 + 2x4 + x2 + 1 p3(x) = x 6 + x5 + x4 + x3 + x2 + x + 1 p4(x) = x 4 + x3 + x2 + x + 1 p5(x) = x + 2. Let the dimension k = 17 . Then the degree of the polynomials g(x) has to be 18. Taking the product of two of the polynomials above, one of degree 12 and one of degree 6, we obtain two polynomials g(x) of degree 18. We choose g(x) = x18 + x17 + 2x16 + 2x15 + x14 + 2x13 + 2x12 + 2x11 + x10 + x9 + 2x4 + 2x2 + 1, and then we search for f2(x). The polynomial f2(x) = x9 + 2x8 + 2x7 + x6 + x4 + x3 + 1 yields a [70,17,29]3 quasi-cyclic code. Afterwards, we search for f3(x) and f4(x) in succession. The polynomial f3(x) = x 10 +x8 +2x6 +x5 +x3 +2x+2 leads to a [105,17,48]3 code and f4(x) = x8 +2x4 +2x3 +x2 +2 gives a [140,17,69]3 code. In the following theorems the defining polynomials are listed with the lowest degree coefficient on the left. 229 http://orcid.org/0000-0001-7441-4757 http://orcid.org/0000-0002-7350-4061 R. Daskalov, P. Hristov / J. Algebra Comb. Discrete Appl. 4(3) (2017) 227–234 Theorem 3.2. There exist one-generator quasi-cyclic codes with parameters: [160,12,90]3, [104,13,54]3, [120,13,62]3, [156,13,86]3, [182,13,102]3, [224,13,127]3, [160,14,87]3, [48,15,18]3, [160,16,84]3, [52,17,19]3, [105,17,48]3, [123,17,59]3, [140,17,69]3, [111,19,49]3, [104,20,45]3 [170,20,82]3. Proof. The coefficients of the defining polynomials of the codes are as follows: A [160,12,90]3 code (m = 80,p = 2): 21021012201011010101000101121011012110020122220021112210200212100111100000000000, 11202022111110110221012101101122200120220001001120211021000110020101102012110000; A [104,13,54]3 code (m = 26,p = 4): 22000102100211000000000000, 22212202222121122101000000, 10221201011011101102210000, 21122112010222002201000000; A [120,13,62]3 code (m = 40,p = 3): 1102201002011122212010121021000000000000, 2001110011212210020202220120121210000000, 2211012011110100012021202001210120010000; A [156,13,86]3 code (m = 52,p = 3): 2012002102201121122202110200200022200021000000000000, 1111211022101012120102220001022012010102022021000000, 1021020221112201002220221102211002022222012200010000; A [182,13,102]3 code (m = 91,p = 2): 12121202201122111220201000101011000200222002000110101000102022111221102202121210 00000000000, 12011200200212120120011020020222110202200002101112002122011221020211201021222100 20112100000; A [224,13,127]3 code (m = 56,p = 4): 22120222120002200111020212101101012000020221000000000000, 21100120002100102020210110221201212000102201100210100000, 22102211102110121220010210100222200112210222112012010000, 12120121221021102201021101002021211201202002222112101000; A [160,14,87]3 code (m = 80,p = 2): 11010200020021110210121201112212021120120202011111222020121210112010000000000000, 21211111201022111210011200101201021111222022210201212111211121022102200100100000; A [48,15,18]3 code (m = 16,p = 3): 2100000000000000, 2022201212010000, 2011012221000000; A [160,16,84]3 code (m = 80,p = 2): 21122202101002000221000021202201102222121011212120112222222221111000000000000000, 11212010202020100022102121021011100212100111210201122001110012101112012110000000; A [52,17,19]3 code (m = 26,p = 2): 21101001110000000000000000, 22120102122100222001210000; A [105,17,48]3 code (m = 35,p = 3): 10202000011222122110000000000000000, 10210201001210110110202100010000000, 22121121212010222210222120011000000; A [123,17,59]3 code (m = 41,p = 3): 10111221201020102122111010000000000000000, 21012100122202021022002102122101000000000, 10001020002202222112112100212221100100000; A [140,17,69]3 code (m = 35,p = 4): 10202000011222122110000000000000000, 10210201001210110110202100010000000, 22112002222210021120212112100000000, 20021202112020221122002110000100000; A [111,19,49]3 code (m = 37,p = 3): 1020220100010220201000000000000000000, 2110000221222002121020122210000000000, 2122202011110212101012201121000000000; A [104,20,45]3 code (m = 52,p = 2): 2001111020002211212012002112211010000000000000000000, 2100222212021021100112122012201100020010110000000000; A [170,20,82]3 code (m = 85,p = 2): 230 http://orcid.org/0000-0001-7441-4757 http://orcid.org/0000-0002-7350-4061 R. Daskalov, P. Hristov / J. Algebra Comb. Discrete Appl. 4(3) (2017) 227–234 2202110000022200100210120122220000112001100102212020020200020100010000000000000000000, 1111002221101112100111212110102011001121001001222102000022221210102102000110000000000. 4. The new codes Let us look at an example, related to the next Theorem 4.1. We will show how the [151,17,75]3 code C is constructed. A generator matrix of a code C has the form G =   G2 0 ∗ G3   , where G2 and G3 are generator matrices of the codes C2 and C3 respectively, and (∗) denotes l linearly independent codewords of a code C1. The generator matrix G2 of the [140,12,75]3 subcode C2 has the same first row as the row given in Theorem 4.1. The generator matrix of the [11,5,6]3 code is G3 =   10122002010 01012200201 10101220020 01010122002 20101012200   and the five independent codewords from the [140,17,69]3 code C1 are: 1020200001122212211000000000000000010210201001210110110202100010000000221120022222100211202 1211210000000020021202112020221122002110000100000, 0102020000112221221100000000000000001021020100121011011020210001000000022112002222210021120 2121121000000002002120211202022112200211000010000, 0010202000011222122110000000000000000102102010012101101102021000100000002211200222221002112 0212112100000000200212021120202211220021100001000, 0001020200001122212211000000000000000010210201001210110110202100010000000221120022222100211 2021211210000000020021202112020221122002110000100, 0000102020000112221221100000000000000001021020100121011011020210001000000022112002222210021 1202121121000000002002120211202022112200211000010; The weight enumerator of the [151,17,75]3 code is 01 75784 762664 787728 7920870 8140824 82111524 84175574 85491414 87610190 881651468 901659156 914416804 933644426 949032618 966278202 9714388718 998407602 10017626234 1028667680 10316568328 1056850326 10611875360 1084084836 1096379366 1111839922 1122579332 114607642 115760172 117143598 118162324 12024976 12123782 1233150 1242198 12684 127266 13212 1358. Theorem 4.1. There exist new ternary linear codes, having parameters: [168,12,96]3, [110,13,57]3, [114,13,60]3, [122,13,64]3, [157,13,87]3, [184,13,104]3, [225,13,128]3, [170,14,93]3, [49,15,19]3, [165,16,86]3, [53,17,20]3, [106,17,49]3, [124,17,60]3, [151,17,75]3, [112,19,50]3, [110,20,48]3, [171,20,83]3. Proof. All of the codes are obtained by construction X. The parameters of the codes, related to Theorem 2.1, are given in Table 1. The defining polynomials of the subcodes C2 are given for clearness. The coefficients of these polynomials are as follows: A [160,10,96]3 code: 21201222020021120202010102102221122201120020112221020002002211221112211000000000, 11011012002221211020222222112100122121122201011101110101210111120002110002201100; A [104,10,57]3 code: 21002111121000201000000000, 21200212021001111212221000, 11121000000010111002211110, 20011221200122201101021000; A [104,9,60]3 code: 231 http://orcid.org/0000-0001-7441-4757 http://orcid.org/0000-0002-7350-4061 R. Daskalov, P. Hristov / J. Algebra Comb. Discrete Appl. 4(3) (2017) 227–234 Table 1. The new codes. code C1 subcode C2 code C3 new code C [160,12,90]3 [160,10,96]3 [8,2,6]3 [168,12,96]3 [104,13,54]3 [104,10,57]3 [6,3,3]3 [110,13,57]3 [104,13,54]3 [104,9,60]3 [10,4,6]3 [114,13,60]3 [120,13,62]3 [120,12,64]3 [2,1,2]3 [122,13,64]3 [156,13,86]3 [156,12,87]3 [1,1,1]3 [157,13,87]3 [182,13,102]3 [182,12,104]3 [2,1,2]3 [184,13,104]3 [224,13,127]3 [224,12,128]3 [1,1,1]3 [225,13,128]3 [160,14,87]3 [160,10,93]3 [10,4,6]3 [170,14,93]3 [48,15,18]3 [48,14,19]3 [1,1,1]3 [49,15,19]3 [160,16,84]3 [160,12,86]3 [5,4,2]3 [165,16,86]3 [52,17,19]3 [52,16,20]3 [1,1,1]3 [53,17,20]3 [105,17,48] 3 [105,16,49]3 [1,1,1]3 [106,17,49]3 [123,17,59]3 [123,16,60]3 [1,1,1]3 [124,17,60]3 [140,17,69]3 [140,12,75]3 [11,5,6]3 [151,17,75] 3 [111,19,49]3 [111,18,50]3 [1,1,1]3 [112,19,50]3 [104,20,45]3 [104,17,48]3 [6,3,3]3 [110,20,48]3 [170,20,82]3 [170,19,83]3 [1,1,1]3 [171,20,83]3 21012011002110011100000000, 21210221012001202202000100, 20102202120100012001122212, 21211221200221021211002002; A [120,12,64]3 code: 1022011202110010021112112222200000000000, 2101002010121022021212001111112122000000, 2020211110002120011122112101122111012000; A [156,12,87]3 code: 2012002102201121122202110200200022200021000000000000, 1111211022101012120102220001022012010102022021000000, 1021020221112201002220221102211002022222012200010000; A [182,12,104]3 code: 11212112011010200101211200121210200210200102100102121200122120200102022012212 12200000000000, 11110110210221211111010221021200202212010002221001102210110102221220111222100 22021101220000; A [224,12,128]3 code: 20211200211002010100221221221021211100021202200000000000, 22020111002220122121222102202111121100122011020222120000, 20222020022202112101012222120200010101022200201111112000, 11211112102222022011222021202122120111112102000201221200; A [160,10,93]3 code: 11122201000222022201201101000220220102212122011021220121120110122122202000000000, 21121101202212200111020011012200211220220120200112201111112111120221012211121020; A [48,14,19]3 code: 2010000000000000,2221121000211000,2212110110100000; A [160,12,86]3 code: 22202221200021221101211122002221022000100211020211121021222221212010100000000000, 10220111202202200200120011220201221200120211101221110011212112001121200212201000; A [52,16,20]3 code: 22021201002000000000000000, 20211122210220200101122000; A [105,16,49]3 code: 232 http://orcid.org/0000-0001-7441-4757 http://orcid.org/0000-0002-7350-4061 R. Daskalov, P. Hristov / J. Algebra Comb. Discrete Appl. 4(3) (2017) 227–234 12212100010100210202000000000000000, 12222211201122102102212220012000000, 20212012121112200022200211010200000; A [123,16,60]3 code: 12100102111221122210200212000000000000000, 22211220110012122220102222210221200000000, 12001221002012000201201220221002020120000; A [140,12,75]3 code: 10202201021222010222112200000000000, 10210102202200101012101001022000200, 22112111102221102020001210012212000, 20021002200002101220121220012000002; A [111,18,50]3 code: 1221201120012201211200000000000000000, 2202000202100102212221110022000000000, 2210012110002221221211011012200000000; A [104,17,48]3 code 2202221011012100122001220021020200020000000000000000, 2011111211011102222020222121112012222021111220000000; A [170,19,83]3 code: 2110101121021000201010101101102012121111122020201112111111212000121111000000000000000, 1201101201202112021100201112201011120012211212121211202221020020211112202022221000000. Theorem 4.2. There exist new ternary linear codes with parameters: [39,12,18]3, [48,13,21]3, [66,18,27]3 and [106,14,54]3. Proof. 1. There exist a [36,12,15]3 QC code C, having the following defining polynomials: 112122201101, 222010220221, 222001010001. This code is triple extendible. By adding the next three columns (110110110110)>, (101101101101)>, (011011011011)> to the generator matrix of C, we get a new self-orthogonal optimal [39,12,18]3 code. The weight distribution of the new code is 01 188034 2148204 24169415 27204464 3090090 3310998 36234. The shortened [38,11,18]3 code is also optimal. 2. There exist a [48,12,21]3 QC code C, having the following defining polynomials: 102100112112, 102121001210, 111002112002, 111210200210. By adding the next row 000000000000000000000000111111111111222222222222 to the generator matrix of code C, we get a new [48,13,21]3 code. 3. First a new [64,18,25]3 code D of type (4) has been constructed. It has defining polyno- mials d1(x) = g(x) = (x8 + 2x4 + 2)(x4 + 2x2 + 2)(x2 + 2x + 2) and d2(x) = g(x).f2(x), where f2(x) = x 13 + x10 + 2x8 + x7 + 2x6 + x5 + x4 + 2x3 + 2x2 + 2x + 1, i.e. the defining polynomials are 21011021012211100000000000000000, 22011120101001120212221001110000. This code is double ex- tendible. By adding the next columns (101010101010101010)>, (01010101010101010)> to the generator matrix of D we get a new [66,18,27]3 code. 4. A new [106,14,54]3 code has been constructed by Construction XX (Theorem 2.2), where C is a [104,14,52]3 code, C1 is a [104,13,53]3 code, C2 is a [104,13,53]3 code, C1 ⋂ C2 is a [104,12,54]3 code and a1 = a2 = 1, k1 = k2 = 1, δ1 = δ2 = 1. The defining polynomials of C, C1, C2 and C1 ⋂ C2 are as follows: 12000020022110000000000000, 22212202200212210221000000, 20122211021022101000000000, 20101021210012001222101100; 11100021020202000000000000, 20021012010221022202200000, 21110020222220221200000000, 21121222122011101100221020; 10200022021021000000000000, 21100122120200101210100000, 22101102120121011100000000, 22111120001010201011011210; 12210020122222200000000000, 22020110211210121122120000, 20221022211112210020000000, 20200011001212211210210122; A generator matrix of the new code has the next form G =   G ? 0 ∗ G3   , where G? is a generator 233 http://orcid.org/0000-0001-7441-4757 http://orcid.org/0000-0002-7350-4061 R. Daskalov, P. Hristov / J. Algebra Comb. Discrete Appl. 4(3) (2017) 227–234 matrix of C1 ⋂ C2, G3 = ( 1 1 1 2 ) and ∗ denotes the next two linearly independent codewords of a code C : 1200002002211000000000000022212202200212210221000000201222110210221010000000002010102121001 2001222101100, 0120000200221100000000000002221220220021221022100000020122211021022101000000000201010212100 1200122210110 . References [1] N. Aydin, I. Siap, D. Ray-Chaudhuri, The structure of 1–generator quasi–twisted codes and new linear codes, Des. Codes Cryptogr. 24(3) (2001) 313–326. [2] A. E. Brouwer, Bounds on the Size of Linear Codes, in Handbook of Coding Theory, V.S. PLess, W.C. Huffman, R.A. Brualdi(eds), Elsevier Amsterdam, 1998. [3] E. Z. Chen, Database of quasi–twisted codes, available at http://www.tec.hkr.se/∼chen/ re- search/codes/searchqc2.htm [4] E. Z. Chen, A new iterative computer search algorithm for good quasi–twisted codes, Des. Codes Cryptogr. 76(2) (2015) 307–323. [5] E. Chen, N. 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Theory 43(6) (1997) 1757–1766. 234 http://orcid.org/0000-0001-7441-4757 http://orcid.org/0000-0002-7350-4061 http://dx.doi.org/10.1023/A:1011283523000 http://dx.doi.org/10.1023/A:1011283523000 http://www.tec.hkr.se/~chen/research/codes/searchqc2.htm http://www.tec.hkr.se/~chen/research/codes/searchqc2.htm http://dx.doi.org/10.1007/s10623-014-9950-8 http://dx.doi.org/10.1007/s10623-014-9950-8 http://dx.doi.org/10.13069/jacodesmath.36947 http://dx.doi.org/10.13069/jacodesmath.36947 http://dx.doi.org/10.13069/jacodesmath.36947 http://dx.doi.org/10.1080/02522667.2014.961788 http://dx.doi.org/10.1080/02522667.2014.961788 http://dx.doi.org/10.1109/18.623167 http://dx.doi.org/10.1109/18.623167 http://dx.doi.org/10.1023/A:1020094206873 http://dx.doi.org/10.1023/A:1020094206873 https://doi.org/10.1109/TIT.2003.815798 https://doi.org/10.1109/TIT.2003.815798 http://dx.doi.org/10.1016/S0012-365X(03)00126-2 http://dx.doi.org/10.1016/S0012-365X(03)00126-2 http://www.codetables.de http://dx.doi.org/10.1007/BF00124211 http://dx.doi.org/10.1007/BF00124211 https://doi.org/10.1109/18.605613 https://doi.org/10.1109/18.605613 https://doi.org/10.1109/18.850694 https://doi.org/10.1109/18.850694 https://doi.org/10.1109/18.641542 https://doi.org/10.1109/18.641542 Introduction Quasi-cyclic codes Good QC codes The new codes References