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Eight Error Correction for (57,29,17) Quadratic Residue Code Over Binary Field 

 

P. Shakila Banua, R.Shobanab 
aAssistant Professor, Department of Mathematics, Vellalar College for Women, 

Erode-638012, Tamilnadu, India 
bResearch Scholar, Department of Mathematics, Vellalar College for Women, 

Erode-638012, Tamilnadu, India 

E-Mail: a shakimeeran10@gmail.com, 
b shobanarj99@gmail.com 

 

 

Article History: 

Received: 26-03-2024 

Revised: 18-05-2024 

Accepted: 30-05-2024 

Abstract:  

This paper introduces novel parameters and presents a method- ology for identifying the 

necessary syndrome indices required to compute the unknown syndromes within the 

context of the (57, 29, 17) quadratic residue code. By determining the resulting index sets, 

the unknown syndromes can be computed, subsequently leading to the derivation of the 

corresponding error-locator polynomial through the application of a de- coding algorithm. 

Keywords: Algebraic Decoding, Quadratic residue code, Index set, Syndromes. 

2020 subject classifications: 94B15, 94B35, 94B60. 

 

1. Introduction 

In 1958, Prange [11] pioneered the quadratic residue (QR) codes. Hamming addressed the issue 

of rectifying a single corrupted binary digit within any sequence of length n during the 

transmission of binary data over a noisy channel. Shapiro H.S. and Slotnick D.L. researched the 

equivalent problem for channels capable of corrupting a larger number of digits [13]. 

MacWilliams F.J. and Sloane N.J.A. comprehensively elucidated various forms of Error-

Correcting Codes and decoding algorithms [7]. M. Elia achieved the decoding of the (23,12,7) 

Golay code by employing the algebraic decoding method for three-error-correcting BCH 

codes [2]. The researchers obtained the results of mathematical and analytical computations for 

multiple binary QR codes [17]. Additionally, they proposed a rapid method for identifying 

primitive polynomials over binary fields in [8]. The extended QR codes of lengths 32 and 48 

exhibit non-linear binary pat- terns with significantly higher minimum weights were discussed in 

[10]. In the decoding process of the (47,24,11) QR code, the author devised two method- ologies to 

ascertain the nonlinear correlations between known and unknown syndromes, effectively rectifying 

five errors and diagnosing six errors [12]. The QR code was generated using the Truong et al 

decoding scheme with parameters (71,36,11), (79,40,15), and (97,49,15), accompanied by 

comprehensive computational modeling [18]. Furthermore, Chen et al. demonstrated a novel 

algebraic decoding technique for the (73,37,13) binary quadratic residue code in [1]. Lin et al. 

developed an amended decoding method specifically tailored for decoding the (48,24,12) 

extended binary QR code. This method enables the correction of up to six errors, leveraging the 

reliability-search algorithm pro- posed by Dubney et al. [16].  Utilizing Newton’s identities, 

the coefficients of the error locator polynomial are determined, facilitating the creation of a 

decoding method aimed at reducing decoding time [15]. Additionally, various enhanced 



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decoding techniques for 11 QR codes have been introduced through Grobner foundation 

techniques [5]. 

AH.P. Lee and H.C. Chang enhanced an algebraic decoding algorithm (ADA) to effectively 

decode up to five errors in binary systematic QR codes [3].  

Additionally, the author proposed a hybrid algebraic decoding algorithm tailored to the 

specified parameters. In cases where the total number of errors v is five or fewer, the Laplace 

formula was utilized to derive the primary unknown syndromes. When v 6 and Gaussian 

elimination is utilized to figure out the unknown syndromes [6]. Truong et al. devised a method 

to compute unknown syndromes for the (73,37,13) QR code, with a focus on enhancing 

decoding performance using soft decisions. A comprehensive study was conducted to evaluate 

error-rate performance [4]. Furthermore, the author elaborated on numerous decoding algorithm 

techniques and error correction coding utilizing a mathematical approach [14]. In a separate 

study, Zhang et al. researched a hard decision (HD) strategy to correct up to five errors and 

decode the (47,24,11) QR code more efficiently [9]. Through simulation results, Zhang et al. 

demonstrated that the new HD algorithm reduces decoding complexity and conserves memory 

while maintaining the same error-rate performance [19]. 

In this paper, Section 1 contains the introduction and Section 2 carry preliminaries. In Section 3, 

the background of the QR code is discussed. The decoding algorithm for the (57,29,17) and the 

unknown syndromes are determined in Section 4. Section 5 contains the application of the 

algorithm and finally, the conclusion for this paper is given in Section 6. 

 

2. Preliminaries 

We will step over some fundamental definitions in this section that are related to our main 

concept. 

 

Definition 2.1. [14] An (𝑛, 𝑘) block code 𝐶 is said to be 𝑐𝑦𝑐𝑙𝑖𝑐 if it is linear and if every 

codeword  𝑐 = (𝑐0, 𝑐1,… . 𝑐𝑛−1) in 𝐶,  its  right cyclic shift  𝑐 , = (𝑐𝑛−1, 𝑐0, … , 𝑐𝑛−2) is also in 

C. 

 

Definition 2.2. [7] Let 𝐺𝐹(𝑙)[𝑥] / (𝑥𝑝  −  1) be a ring, where a prime number is p and the 

quadratic residue of 𝑝 is 𝑙, (𝑥𝑝  −  1) = (𝑥 −  1)𝑞(𝑥)𝑛(𝑥). Define the set of quadratic 

residues modulo 𝑝 by 𝑄𝑝, and the set of quadratic non-residues by 𝑁𝑝, 𝑄, 𝑄
,, 𝑁 and 𝑁′are 

quadratic residue codes, which are cyclic cod es  (or ideals) of the ring with generator 

polynomials of 𝑞(𝑥), (𝑥 − 1)𝑞(𝑥), 𝑛(𝑥),  (𝑥 − 1)𝑛(𝑥), so forth , where  𝑞(𝑥) =

∏ (𝑥 − 𝛼𝑖)𝑖∈𝑄𝑝 , 𝑞(𝑥) = ∏ (𝑥 − 𝛼𝑖)𝑖∈𝑄𝑝  have coefficients from GF(l). In a field that contains 

GF(l), α represents a primitive 𝑝𝑡ℎ root of unity. 

 

Definition 2.3. [14] An (n, k) cyclic code has a unique minimal monic polynomial 𝑔(𝑥), which 

is the generator of the ideal. This is called the generator polynomial for the code. Let the degree 

of 𝑔 be 𝑛 –  𝑘,  𝑔(𝑥) = 𝑔0 + 𝑔1𝑥 +⋯+ 𝑔𝑛−𝑘𝑥
𝑛−𝑘. 

 



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Definition 2.4. [14] An irreducible polynomial 𝑝(𝑥) ∈ 𝐺𝐹(𝑝)[𝑥] of degree 𝑚 is said to be primitive 

if the smallest positive integers 𝑛 for which 𝑝(𝑥) divides 𝑥𝑛 − 1 is 𝑛 = 𝑝𝑚 − 1. 

 

Definition 2.5. [14] A sequence generated by a connection polynomial 𝑔(𝑥) of degree 𝑛 is said 

to be a maximal length sequence if the period of the sequence is 2𝑛 − 1. 

 

Definition 2.6. [14] A connection polynomial which produces a maximal- length sequence 

is a primitive polynomial. 

3. Non-Binary (57,29,17) QR Code 

The (n, k, d) parameters provide essential information about the capabilities and performance of 

error-correcting codes, guiding their design, implementation, and usage in various 

communication and storage systems. Let (𝑛,
𝑛+1

2
, 𝑑) represent a binary QR code with generator 

polynomial 𝑔(𝑥) over 𝐺𝐹(2). The code length 𝑛, should be a prime number of the form 𝑛 =

8𝑙 ± 1, where 𝑚 is the smallest positive integer such that 𝑛 divides 2𝑚 − 1 and 1 is an arbitrary 

positive integer. The set 𝑄 of quadratic residue modulo 𝑛 is the set of nonzero squares modulo 𝑛 

that is, 

𝑄𝑛 = {𝑗|𝑗 ≡ 𝑥
2 𝑚𝑜𝑑 𝑛, 1 ≤ 𝑥 ≤ 𝑛 − 1}  (1) 

 

For the binary (57, 29, 17) QR code, the components of codeword are in finite field GF (228) 

and its quadratic residue set is 

 

𝑄57 = {1,4,6,7,9,11,16,19,21,24,25,28,30,31,36,39,41,42,43,45,49,51,54,55}  (2) 

 

A root of the primitive polynomial 𝑥28  + 𝑥3  +  1 should be 𝛼 ∈ 𝐺𝐹(228) [1]. The 

multiplicative group of nonzero elements in the finite field 𝐺𝐹(228) is thus generated by 𝛼. It 

follows that a primitive 57𝑡ℎ root of unity is 𝛽 =  𝛼𝑢. where 𝑢 = (228 − 1)/57 =

 4, 709. The generator polynomial 𝑔(𝑥) is defined by, 

 

𝑔(𝑥) =∏ (𝑥 − 𝛽𝑖)
𝑖∈𝑄57

 

 

= 𝑥28 + 𝑥26 + 𝑥24 + 𝑥23 + 𝑥22 + 𝑥21 + 𝑥19 + 𝑥15 + 𝑥14 + 𝑥13 + 𝑥12 + 𝑥7 + 𝑥6 + 𝑥5 + 𝑥3 + 𝑥2

+ 𝑥 + 1 

 

where the multiplicative order of the integer 2 modulo the code length 𝑛 =  57 is represented 

by the degree of g(x) which is 28.  So, 228 ≡ 1(𝑚𝑜𝑑 57) and where 𝑔(𝛽)  =  0. An error 

pattern is considered correctable for the (57, 29, 17) QR Code if its weight is less than or 

equal to the error-correcting capacity 𝑡 =
17−1

2
= 8. Let us now consider a noisy 

channel and the codeword 

 



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𝑐(𝑥) = 𝑐0 + 𝑐1𝑥 + 𝑐2𝑥
2 +⋯+ 𝑐56𝑥

56 

 

𝑒(𝑥) = 𝑒0 + 𝑒1𝑥 + 𝑒2𝑥
2 +⋯+ 𝑒56𝑥

56 

 

𝑟(𝑥) = 𝑟0 + 𝑟1𝑥 + 𝑟2𝑥
2 +⋯+ 𝑟56𝑥

56 

 

respectively, correspond to the error pattern and the received vector. Next, the word that 

was received takes the form 𝑟(𝑥)  =  𝑐(𝑥) +  𝑒(𝑥). Define 𝑆𝑖 = 𝑟(𝛽
𝑖) = 𝑒(𝛽𝑖), 𝑖 ∈

𝑄57 as the set of known syndromes that can be immediately computed by evaluating 𝑟(𝑥) at 

the roots of 𝑔(𝑥). These syndromes are referred to be unknown syndromes if 𝑖 is absent from 

the set 𝑄57. It is said that the syndrome 𝑠𝑖 is a known syndrome if 𝑖 ∈ 𝑄57. If not, it’s 

referred to as an unidentified syndrome. The unidentified syndromes are discovered in (2). 

There is a relationship between the syndromes and the QR code, 𝑆2𝑖 = 𝑆𝑖
2, with indices 

modulo 𝑛. such as, 𝑆2 = 𝑆1
2, 𝑆4 = 𝑆2

2 = 𝑆1
4, 𝑆8 = 𝑆4

2 = 𝑆1
8, 𝑆16 = 𝑆8

2 = 𝑆1
16, 𝑆64 = 𝑆32

2 =

𝑆1
64, 𝑆128 = 𝑆64

2 = 𝑆1
128, 𝑆256 = 𝑆128

2 = 𝑆1
256. The error locator patterns is defined by, 

 

𝐿(𝑧) = ∏ (𝑧 − 𝑧𝑖) = 𝑧
𝑣 + ∑ 𝜎𝑗𝑧

𝑣−𝑗𝑣
𝑗=1

𝑣
𝑖=1   (3) 

 

𝜎1 = 𝑧1 + 𝑧2 +⋯+ 𝑧𝑣 

𝜎2 = 𝑧1𝑧2 + 𝑧1𝑧3…𝑧𝑣−1𝑧𝑣 

𝜎3 = 𝑧2𝑧3 + 𝑧2𝑧4…𝑧2𝑣 

. 

. 

. 

𝜎𝑣 = 𝑧1𝑧2…𝑧𝑣 

 

Thus, by using the Chien search algorithms to locate the error 𝑧1𝑧2…𝑧𝑣 as roots of 

polynomial 𝐿(𝑧) in (3), it is easy to find the elementary symmetric functions 𝜎𝑖.The 

coefficients of 𝐿(𝑧) are found using the following Newton Identities: 

 

𝑠1 + 𝜎1 = 0 

𝑠2 + 𝜎1𝑠1 + 2𝜎2 = 0 

𝑠3 + 𝜎1𝑠2 + 𝜎2𝑠1 + 3𝜎3 = 0 

. 

. 

. 

𝑠𝑣 + 𝜎1𝑠𝑣−1 +…+ 𝜎𝑣−1𝑠1 + 𝑣𝜎𝑣 = 0 

 

The decoding algorithm is used to decode the QR code up to 8 errors. The 𝑆1, 𝑆2, … , 𝑆16 

syndromes are the first 16 in succession. However, only the syndromes S1, S4, S6, S7, S9, S11, 

S16 can be calculated directly from 𝑟(𝑥) and the others S2, S3, S5, S8, S10, S12, S13, S14, S15 are 



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not determined directly from 𝑟(𝑥), which can be expressed in powers of S2 and S15. 

 

4. Decoding algorithm for (57, 29, 17) QR code 

One can apply the decoding algorithm to the receive data bits to recover the original error. A 

new decoding algorithm for the code is given below. 

 

Step 1:  The first 16 consecutive syndromes S1, S2……., S16. 

Step 2:  Obtain the known syndromes S1, S4, S6, S7, S9, S11, S16. 

Step 3: If odd syndromes are zero, (i.e.) S1 = S7 = S9 = S11 = 0, assume no errors 

occur and stop. 

Step 4: Choose the unknown syndromes and set v =1. 

Step 5: Choose a subset 𝐼 = 𝑖1, 𝑖2, … , 𝑖𝑣+1  ⊂ 𝑄. 

Step 6: Choose a subset J containing v + 1 elements from the difference set in Step 4. If all 

the possible sets J have been returning to Step 2. 

Step 7: If the intersection of the multi-set 𝐼⊕ 𝐽 is empty, return to Step 4. 

Step 8: Computing the consecutive unknown syndromes for possible pair 𝑆2
𝑣, 𝑆15

𝑣 . 

Step 9: If there exists one monomial of the unknown syndrome whose coefficient is 1 and 

whose power is different from that of other monomials, then stop; otherwise, return to Step 4. 

 

4.1 Determination of Unknown syndromes 

Assume that v errors occur in the received word.  Let I = i1, i2, ...iv+1 and J = j1, j2, ...jv+1 

denote two subsets of 0,1, 2...56, respectively.  Next, consider the matrix (I, J) of size (v + 1) 

× (v + 1) given by, 

 

𝑆(𝐼, 𝐽) = [

𝑆𝑖1+𝑗1 ⋯ 𝑆𝑖1+𝑗𝑣+1
⋮ ⋱ ⋮

𝑆𝑖𝑣+1+𝑗1 ⋯ 𝑆𝑖𝑣+1+𝑗𝑣+1

] 

 

where the summation of the indices of the is modulo 𝑛 and the rank of  𝑆(𝐼, 𝐽) is at most 𝑣, 

which in turn implies the following equation,  

𝑑𝑒𝑡 𝑆(𝐼, 𝐽)  =  0  (4) 

 

Now, assuming the subsets 𝐼and 𝐽 for the code, 

 

𝐼 ⊕ 𝐽 = {(𝑖 + 𝑗) 𝑚𝑜𝑑 57|𝑖 ∈ 𝐼, 𝑗 ∈ 𝐽} 

 

Example: The sum of two subsets 𝐼and 𝐽 are illustrated below. 

If 𝑣 = 5, 𝐼 = {0,1,2,3,4} and 𝐽 = {1,2,3,4,5}, then 

 

𝐼 ⊕ 𝐽 = {0 + 1, 0 + 2, 0 + 3, 0 + 4, 0 + 5, 1 + 1, 1 + 2, 1 + 3, 1 + 4, 1 + 5, 2 + 1, 2 + 2, 2 + 3, 2

+ 4, 2 + 5, 3 + 1, 3 + 2, 3 + 3, 3 + 4, 3 + 5, 4 + 1, 4 + 2,+4 + 3, 4 + 4, 4 + 5 

 



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𝐼 ⊕ 𝐽 = {1, 2, 2, 3, 3, 3, 4, 4, 4, 4, 5, 5, 5, 5, 5, 6, 6, 6, 6, 7, 7, 7, 8, 8, 9} 

 

It was proposed to find the corresponding subsets I and J. The following steps are involved in 

the determination of unknown syndromes, 

 

Case 0: (zero error) No error in the received codeword if 𝑆1 = 0; Otherwise go to case 1. 

 

Case 1: (One error) 𝑆2 = 𝑆1
2, 𝑆15 = 𝑆1

15 

 

Case 2: (Two errors) Let 𝐼1 = {0,4,1}, 𝐽1 = {7,0,1}, 𝐼2 = {0,4,7},  𝐽2 = {3,0,8}. The 

matrices are, 

𝑆(𝐼1, 𝐽1) = (
𝑆7 𝑆0 𝑆1
𝑆11 𝑆4 𝑺𝟓
𝑆4 𝑆1 𝑺𝟐

)   

 

𝑆(𝐼2, 𝐽2) = (
𝑆3 𝑆0 𝑆8
𝑆7 𝑆4 S12
𝑆10 𝑆7 𝑺𝟏𝟓

) 

 

The corresponding monomial in 𝑑𝑒𝑡 (𝑆(𝐼1, 𝐽1)) (𝑑𝑒𝑡 (𝑆(𝐼2, 𝐽2)))  𝑖𝑠 𝑆2
1, 𝑆15

1 . 

 

Case 3: (Three errors) Let 𝐼1 = {7,0,1,2}, 𝐽1 = {9,2,1,0}, 𝐼2 = {11,7,8,10},  𝐽2 =

{9,8,7,5}. The matrices are, 

𝑆(𝐼1, 𝐽1) = (

𝑆16 𝑆9 𝑺𝟖 𝑆7
𝑆9 𝑺𝟐 𝑆1 𝑆0
𝑺𝟏𝟎 𝑺𝟑 𝑺𝟐 𝑆1
𝑆11 𝑆4 𝑺𝟑 𝑺𝟐

) 

 

𝑆(𝐼2, 𝐽2) = (

𝑺𝟐𝟎 𝑆19 𝑺𝟏𝟖 𝑆16
𝑆16 𝑺𝟏𝟓 𝑺𝟏𝟒 𝑺𝟏𝟐
𝑺𝟏𝟕 𝑆16 𝑺𝟏𝟓 𝑺𝟏𝟑
𝑆19 𝑺𝟏𝟖 𝑺𝟏𝟕 𝑺𝟏𝟓

) 

 

The corresponding monomial in 𝑑𝑒𝑡 (𝑆(𝐼1, 𝐽1)) (𝑑𝑒𝑡 (𝑆(𝐼2, 𝐽2)))  𝑖𝑠 𝑆2
3, 𝑆15

3 . 

 

Case 4: (Four errors) Let 𝐼1 = {0,2,1,4,3}, 𝐽1 = {2,0,1,6,9}, 𝐼2 = {0,9,1,15,4},  𝐽2 =

{15,6,14,0,11}. The matrices are, 

 

𝑆(𝐼1, 𝐽1) =

(

 
 

𝑺𝟐 𝑆0 𝑆1 𝑆6 𝑆9
𝑆4 𝑺𝟐 𝑺𝟐 𝑺𝟖 𝑆11
𝑺𝟐 𝑆1 𝑺𝟐 𝑆7 𝑺𝟏𝟎
𝑆6 𝑆4 𝑺𝟓 𝑺𝟏𝟎 𝑺𝟏𝟑
𝑺𝟓 𝑺𝟑 𝑆4 𝑆9 𝑺𝟏𝟐)

 
 

 

 



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𝑆(𝐼2, 𝐽2) =

(

 
 

𝑺𝟏𝟓 𝑆6 𝑺𝟏𝟒 𝑆0 𝑆11
𝑆24 𝑺𝟏𝟓 𝑺𝟐𝟑 𝑆9 𝑺𝟐𝟎
𝑆16 𝑆7 𝑺𝟏𝟓 𝑆1 𝑺𝟏𝟐
𝑆30 𝑆21 𝑺𝟐𝟗 𝑺𝟏𝟓 𝑺𝟐𝟔
𝑆19 𝑺𝟏𝟎 𝑆18 𝑆4 𝑺𝟏𝟓)

 
 

 

 

The corresponding monomial in 𝑑𝑒𝑡 (𝑆(𝐼1, 𝐽1)) (𝑑𝑒𝑡 (𝑆(𝐼2, 𝐽2)))  𝑖𝑠 𝑆2
5, 𝑆15

5 . 

 

Case 5: (Five errors) Let 𝐼1 = {1,0,12,2,20,40}, 𝐽1 = {1,2,13,0,5,3}, 𝐼2 =

{15,6,2,45,32,1},  𝐽2 = {0,9,13,20,29,55}. The matrices are, 

 

𝑆(𝐼1, 𝐽1) =

(

 
 
 

𝑺𝟐 𝑺𝟑 𝑺𝟏𝟒 𝑆1 𝑆6 𝑆4
𝑆1 𝑺𝟐 𝑺𝟏𝟑 𝑆0 𝑺𝟓 𝑺𝟑
𝑺𝟏𝟑 𝑺𝟏𝟒 𝑆25 𝑺𝟏𝟐 𝑺𝟏𝟕 𝑺𝟏𝟓
𝑺𝟑 𝑆4 𝑺𝟏𝟓 𝑺𝟐 𝑆7 𝑺𝟏𝟓
𝑆21 𝑺𝟐𝟐 𝑺𝟑𝟑 𝑺𝟐𝟎 𝑆25 𝑺𝟐𝟑
𝑆41 𝑆42 𝑆53 𝑺𝟒𝟎 𝑆45 𝑆43)

 
 
 

 

 

𝑆(𝐼2, 𝐽2) =

(

 
 
 

𝑺𝟏𝟓 𝑆24 𝑆28 𝑺𝟑𝟓 𝑺𝟒𝟒 𝑺𝟏𝟑
𝑆6 𝑺𝟏𝟓 𝑆19 𝑺𝟐𝟔 𝑺𝟑𝟓 𝑺𝟒
𝑺𝟐 𝑆11 𝑺𝟏𝟓 𝑺𝟐𝟐 𝑆31 𝑆57
𝑆45 𝑆54 𝑆1 𝑺𝟖 𝑺𝟏𝟕 𝑆43
𝑺𝟑𝟐 𝑆41 𝑆45 𝑺𝟓𝟐 𝑆4 𝑆30
𝑆1 𝑺𝟏𝟎 𝑺𝟏𝟒 𝑆21 𝑆30 𝑺𝟓𝟔)

 
 
 

 

 

The corresponding monomial in 𝑑𝑒𝑡 (𝑆(𝐼1, 𝐽1)) (𝑑𝑒𝑡 (𝑆(𝐼2, 𝐽2)))  𝑖𝑠 𝑆2
4, 𝑆15

3 . 

 

Case 6: (Six errors) Let 𝐼1 = {2,0,1,50,47,21,12}, 𝐽1 = {0,2,1,19,16,4,50}, 𝐼2 =

{13,1,0,11,39,51,4},  𝐽2 = {2,14,15,8,45,7,18}. The matrices are, 

 

𝑆(𝐼1, 𝐽1) =

(

 
 
 
 

𝑺𝟐 𝑆4 𝑆3 𝑆21 𝑺𝟏𝟖 𝑆6 𝑆52
𝑆0 𝑺𝟐 𝑆1 𝑆19 𝑆16 𝑆4 𝑺𝟓𝟎
𝑆1 𝑺𝟑 𝑺𝟐 𝑺𝟐𝟎 𝑆17 𝑺𝟓 𝑆51
𝑺𝟓𝟎 𝑺𝟓𝟐 𝑆51 𝑺𝟏𝟐 𝑆9 𝑆54 𝑆43
𝑺𝟒𝟕 𝑆49 𝑺𝟒𝟖 𝑆9 𝑆6 𝑆51 𝑺𝟒𝟎
𝑆21 𝑺𝟐𝟑 𝑺𝟐𝟐 𝑺𝟒𝟎 𝑺𝟑𝟕 𝑆25 𝑺𝟏𝟒
𝑺𝟏𝟐 𝑺𝟏𝟒 𝑺𝟏𝟑 𝑆31 𝑆28 𝑆16 𝑺𝟓)

 
 
 
 

 

 

𝑆(𝐼2, 𝐽2) =

(

 
 
 
 

𝑺𝟏𝟓 𝑺𝟐𝟕 𝑆28 𝑆21 𝑆1 𝑺𝟐𝟎 𝑆31
𝑺𝟑 𝑺𝟏𝟓 𝑆16 𝑆9 𝑺𝟒𝟔 𝑺𝟖 𝑆19
𝑺𝟐 𝑆4 𝑺𝟏𝟓 𝑺𝟖 𝑆45 𝑆7 𝑺𝟏𝟖
𝑺𝟏𝟑 𝑆25 𝑺𝟐𝟔 𝑆19 𝑺𝟓𝟔 𝑺𝟏𝟖 𝑆29
𝑆41 𝑺𝟓𝟑 𝑆54 𝑺𝟒𝟕 𝑺𝟐𝟕 𝑺𝟒𝟔 𝑆57
𝑺𝟓𝟑 𝑺𝟖 𝑆9 𝑺𝟐 𝑆39 𝑺𝟏 𝑺𝟏𝟐
𝑆6 𝑺𝟏𝟖 𝑆19 𝑺𝟏𝟐 𝑆49 𝑆11 𝑺𝟐𝟐)

 
 
 
 

 



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The corresponding monomial in 𝑑𝑒𝑡 (𝑆(𝐼1, 𝐽1)) (𝑑𝑒𝑡 (𝑆(𝐼2, 𝐽2)))  𝑖𝑠 𝑆2
3, 𝑆15

3 . 

 

Case 7: (Seven errors) Let 𝐼1 = {21,41,2,5,0,1,30,7}, 𝐽1 = {4,6,0,30,2,1,11,21}, 𝐼2 =

{1,0,13,11,28,39,55,16},  𝐽2 = {14,15,2,4,24,45,39,0}. The matrices are, 

 

𝑆(𝐼1, 𝐽1) =

(

 
 
 
 
 

𝑆25 𝑺𝟐𝟕 𝑆21 𝑆51 𝑆23 𝑺𝟐𝟐 𝑺𝟑𝟐 𝑆42
𝑆45 𝑺𝟒𝟕 𝑆41 𝑺𝟏𝟒 𝑆43 𝑆42 𝑺𝟓𝟐 𝑺𝟓
𝑆6 𝑺𝟖 𝑺𝟐 𝑺𝟑𝟐 𝑆4 𝑺𝟑 𝑺𝟏𝟑 𝑺𝟐𝟑
𝑆9 𝑆11 𝑺𝟓 𝑺𝟑𝟓 𝑆7 𝑆6 𝑆16 𝑺𝟐𝟔
𝑆4 𝑆6 𝑆0 𝑆30 𝑺𝟐 𝑆1 𝑆11 𝑆21
𝑺𝟓 𝑆7 𝑆1 𝑆31 𝑺𝟑 𝑺𝟐 𝑺𝟏𝟐 𝑺𝟐𝟐
𝑺𝟑𝟒 𝑆36 𝑆30 𝑺𝟑 𝑺𝟑𝟐 𝑆31 𝑆41 𝑆51
𝑆11 𝑺𝟏𝟑 𝑆7 𝑺𝟑𝟕 𝑆9 𝑺𝟖 𝑺𝟏𝟖 𝑆28)

 
 
 
 
 

 

 

𝑆(𝐼2, 𝐽2) =

(

 
 
 
 
 

𝑺𝟏𝟓 𝑆16 𝑺𝟑 𝑺𝟓 𝑆25 𝑺𝟒𝟔 𝑺𝟒𝟎 𝑆1
𝑺𝟏𝟒 𝑺𝟏𝟓 𝑺𝟐 𝑆4 𝑆24 𝑆45 𝑆39 𝑆0
𝑺𝟐𝟕 𝑆28 𝑺𝟏𝟓 𝑺𝟏𝟕 𝑺𝟑𝟕 𝑆1 𝑺𝟓𝟐 𝑺𝟏𝟑
𝑆25 𝑺𝟐𝟔 𝑺𝟏𝟑 𝑺𝟏𝟓 𝑺𝟑𝟓 𝑺𝟓𝟔 𝑺𝟓𝟎 𝑆11
𝑆42 𝑆43 𝑆30 𝑺𝟑𝟐 𝑺𝟓𝟐 𝑆16 𝑺𝟏𝟎 𝑆28
𝑺𝟓𝟑 𝑆54 𝑺𝟒𝟏 𝑆43 𝑆6 𝑆28 𝑆21 𝑆39
𝑺𝟏𝟐 𝑺𝟏𝟑 𝑺𝟓𝟕 𝑺𝟐 𝑺𝟐𝟐 𝑆43 𝑺𝟑𝟕 𝑆55
𝑆30 𝑆31 𝑺𝟏𝟖 𝑆20 𝑺𝟒𝟎 𝑆4 𝑆55 𝑆16)

 
 
 
 
 

 

 

The corresponding monomial in 𝑑𝑒𝑡 (𝑆(𝐼1, 𝐽1)) (𝑑𝑒𝑡 (𝑆(𝐼2, 𝐽2)))  𝑖𝑠 𝑆2
3, 𝑆15

4 . 

 

Case 8: (Eight errors) Let 𝐼1 = {51,0,2,1,8,49,11,21,3}, 𝐽1 = {8,2,0,1,51,4,55,5,3}, 

𝐼2 = {0,1,13,51,28,4,16,9,31},  𝐽2 = {15,14,2,21,11,13,6,1,3}. The matrices are, 

 

𝑆(𝐼1, 𝐽1) =

(

 
 
 
 
 
 

𝑺𝟐 𝑺𝟓𝟑 𝑆51 𝑺𝟓𝟐 𝑆45 𝑆55 𝑆49 𝑺𝟓𝟔 𝑆54
𝑺𝟖 𝑺𝟐 𝑆0 𝑆1 𝑆51 𝑆4 𝑆55 𝑺𝟓 𝑺𝟑
𝑺𝟏𝟎 𝑆4 𝑺𝟐 𝑺𝟑 𝑺𝟓𝟑 𝑆6 𝑆57 𝑆7 𝑺𝟓
𝑆9 𝑺𝟑 𝑆1 𝑺𝟐 𝑺𝟓𝟐 𝑺𝟓 𝑺𝟓𝟔 𝑆6 𝑆4
𝑆9 𝑺𝟏𝟎 𝑺𝟖 𝑆9 𝑺𝟐 𝑆12 𝑆6 𝑺𝟏𝟑 𝑆11
𝑆16 𝑆51 𝑆49 𝑺𝟓𝟎 𝑆43 𝑺𝟓𝟑 𝑺𝟒𝟕 𝑆54 𝑺𝟓𝟐
𝑆57 𝑺𝟏𝟑 𝑆11 𝑆12 𝑆5 𝑆15 𝑆9 𝑆16 𝑆14
𝑺𝟐𝟗 𝑺𝟐𝟑 𝑆21 𝑺𝟐𝟐 𝑺𝟏𝟓 𝑆25 𝑆19 𝑺𝟐𝟔 𝑆24
𝑆11 𝑺𝟓 𝑺𝟑 𝑺𝟑𝟒 𝑆54 𝑆7 𝑆1 𝑺𝟖 𝑆6 )

 
 
 
 
 
 

 

 



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𝑆(𝐼2, 𝐽2) =

(

 
 
 
 
 
 
 

𝑺𝟏𝟓 𝑺𝟏𝟒 𝑺𝟐 𝑆21 𝑆11 𝑺𝟏𝟑 𝑆6 𝑆1 𝑺𝟑
𝑆16 𝑺𝟏𝟓 𝑺𝟑 𝑺𝟐𝟐 𝑺𝟏𝟐 𝑺𝟏𝟒 𝑆7 𝑺𝟐 𝑆4
𝑆28 𝑺𝟐𝟕 𝑺𝟏𝟓 𝑺𝟑𝟒 𝑆24 𝑺𝟐𝟔 𝑆19 𝑺𝟏𝟒 𝑆16
𝑆9 𝑺𝟖 𝑺𝟓𝟑 𝑺𝟏𝟓 𝑺𝟓 𝑆7 𝑆57 𝑺𝟓𝟐 𝑆54
𝑆43 𝑆42 𝑆30 𝑆49 𝑆39 𝑆41 𝑺𝟑𝟒 𝑺𝟐𝟗 𝑆31
𝑆19 𝑺𝟏𝟖 𝑆6 𝑆25 𝑺𝟏𝟓 𝑺𝟏𝟕 𝑺𝟏𝟎 𝑺𝟓 𝑆7
𝑆31 𝑆30 𝑺𝟏𝟖 𝑺𝟑𝟕 𝑺𝟐𝟕 𝑺𝟐𝟗 𝑺𝟐𝟐 𝑺𝟏𝟕 𝑆19
𝑆24 𝑺𝟐𝟑 𝑆11 𝑆30 𝑺𝟐𝟎 𝑺𝟐𝟐 𝑺𝟏𝟓 𝑺𝟏𝟎 𝑺𝟏𝟐
𝑆46 𝑆45 𝑺𝟑𝟑 𝑺𝟓𝟐 𝑆42 𝑺𝟒𝟒 𝑆37 𝑺𝟑𝟐 𝑺𝟑𝟒)

 
 
 
 
 
 
 

 

 

The corresponding monomial in 𝑑𝑒𝑡 (𝑆(𝐼1, 𝐽1)) (𝑑𝑒𝑡 (𝑆(𝐼2, 𝐽2)))  𝑖𝑠 𝑆2
5, 𝑆15

4 . 

 

The above conviction is described in the following flowchart: 

 

5. Application for the algorithm 

The algorithm has been successfully implemented to the provided example, resulting in enhanced 

performance and efficiency. Let (17,9,5) QR code over 𝐺𝐹(28 ) generated by the primitive 



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polynomial 𝑥8 + 𝑥6 + 𝑥5 + 𝑥 + 1. The set 𝑄17  =  {1, 2, 4, 8, 9, 13, 15, 16}. Therefore 𝛽 =  𝛼𝑢 is a 

primitive 17𝑡ℎ root of unity and 𝑢 =  (28  −  1)/17 =  15. This code can correct upto two errors.  

 

For two error cases, 0 ≤  𝑣 ≤  2.  

 

For 𝑣 = 1,  𝐼1 = {1,4}, 𝐽1 = {2,1}. The matrices are,  

 

𝑆(𝐼1, 𝐽1) =  (
𝑺𝟑 𝑆2
𝑆6 𝑆5

) 

 

The corresponding monomial in 𝑑𝑒𝑡 (𝑆(𝐼1, 𝐽1))  𝑖𝑠 𝑆3
1. 

 

For 𝑣 = 2,  𝐼1 = {2,1,8}, 𝐽1 = {4,2,1}. The matrices are,  

𝑆(𝐼1, 𝐽1) = (
𝑆6 𝑆4 𝑺𝟑
𝑆5 𝑺𝟑 𝑆2
𝑆12 𝑆10 𝑆9

) 

 

The corresponding monomial in 𝑑𝑒𝑡 (𝑆(𝐼1, 𝐽1))  𝑖𝑠 𝑆3
2. 

 

Assume the message polynomial 𝐼(𝑥)  =  𝑥5  +  𝑥 +  1 and the code polynomial c(x) = x16 

+ x15 + x13 + x9 + x8 + x4 + x2 + x1 + x + 1. Two error cases are given below. 

 

Case 1: For one error, assume the error polynomial 𝑒(𝑥) = 𝑥3 and the received 

polynomial is, r(x) = x16 + x15 + x13 + x9 + x8 + x5 + x3 + x + 1. Unknown syndrome 

for one error 𝑆3
1 = 𝛼14. The error locator polynomial 𝐿(𝑧)  =  1 + 𝛼18𝑥. The root of 

the σ(x) is 𝑥1 = 𝛼
7 and the error polynomial 𝑒(𝑥) = 𝑥3. 

 

Case 2: For two error, assume the error polynomial 𝑒(𝑥) = 𝑥3+𝑥2 and the received 

polynomial is, r(x) = x16 + x15 + x13 + x9 + x8 + x5 + x3 + x + 1. Unknown syndrome 

for two error 𝑆3
2 = 𝛼16. The error locator polynomial 𝐿(𝑧)  =  1 + 𝛼12𝑥. The root of 

the σ(x) is 𝑥1 = 𝛼
8 and the error polynomial 𝑒(𝑥) = 𝑥3+𝑥2. 

 

6. Conclusion 

In this manuscript, we present an original non-binary quadratic residue code characterized by 

parameters (57, 29, 17), operating within a binary field. Our approach involves the adaptation of 

methods analogous to those employed in discerning unknown syndromes within binary 

quadratic residue codes, tailored for this non-binary code of length 57. By meticulously 

selecting suitable subsets and index sets, we effectively tackle eight instances of errors, 

subsequently resolving them with the aid of a pioneering decoding algorithm. Furthermore, we 

develop into the practical application of this algorithm within our study. 

 

 



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