Binary Quadratic Residue Code Research Articles

Several families of binary cyclic codes with good parameters

Binary odd-like duadic codes have parameters [n,(n+1)/2], where n is odd. It is well known that the minimum odd weight of odd-like duadic codes has the lower bound n. The binary quadratic-residue codes and the punctured binary Reed-Muller codes of certain order are two families of binary cyclic codes with parameters [n,(n+1)/2,d≥n]. It is very hard to construct an infinite family of binary cyclic codes with length n and dimension near (n+1)/2 whose minimum distances have a square-root bound. Recently, Tang and Ding constructed an infinite family of binary cyclic codes with parameters [2m−1,2m−1,d] whose minimum distances have lower bounds close to the square-root bound. The objective of this paper is to construct several infinite families of binary cyclic codes of length 2m−1 and dimension near 2m−1 whose minimum distance d much exceeds the square-root bound. For m≥9 being odd, an infinite family of binary cyclic codes with parameters [2m,2m−1,d≥3⋅2m−12−1] is presented. For m≥8 being even, an infinite family of binary cyclic codes with parameters [2m,2m−1−1+m2,d≥3⋅2m−22+1] is presented.

Efficient Decoding Algorithm for Binary Quadratic Residue Codes Using Reduced Permutation Sets

The Quadratic Residue (QR) codes have a rich mathematic structure. Unfortunately, their Algebraic Decoding (AD) is not generalizable for all QR codes. In this study, an efficient hard decoding algorithm is proposed to generalize the decoding of the binary systematic Quadratic Residue (QR) codes. The proposed decoder corrects t erroneous bits or less, in the received word, based on a reduced set of permutations derived from the large automorphism group of QR codes. This set of permutations is applied to the received word to move the error positions and trap all of them in redundancy. Then, to evaluate the proposed method, we applied it to many binary QR codes of moderate code length starting with 17 until 113 with reducible and irreducible generator polynomials. The proposed decoder was validated by inserting all possible error patterns, that have t or less erroneous positions, as input of the proposed decoder and the output is always a correct codeword. The complexity study, in terms of the number of operations used, reveals that the light permutation decoding LPD algorithm significantly decreases decoding complexity without performance loss. So, it is qualified to be a good competitor to decode QR codes with lower lengths but is the best for QR codes with higher lengths.

Decoding Quadratic Residue Codes Using Deep Neural Networks

In this paper, a low-complexity decoder based on a neural network is proposed to decode binary quadratic residue (QR) codes. The proposed decoder is based on the neural min-sum algorithm and the modified random redundant decoder (mRRD) algorithm. This new method has the same asymptotic time complexity as the min-sum algorithm, which is much lower than the difference on syndromes (DS) algorithm. Simulation results show that the proposed algorithm achieves a gain of more than 0.4 dB when compared to the DS algorithm. Furthermore, a simplified approach based on trapping sets is applied to reduce the complexity of the mRRD. This simplification leads to a slight loss in error performance and a reduction in implementation complexity.

The extended binary quadratic residue code of length 42 holds a 3‐design

AbstractThe codewords of weight 10 of the [42, 21, 10] extended binary quadratic residue code are shown to hold a design of parameters . Its automorphism group is isomorphic to . Its existence can be explained neither by a transitivity argument, nor by the Assmus–Mattson theorem.

Algebraic decoding of the (41, 21, 9) quadratic residue code without determining the unknown syndromes

Quadratic residue (QR) codes have a high prospect on error correction for reliable data conveyance over channel with noise. This paper presents a fast algebraic scheme for decoding the binary (41, 21, 9) QR code for correcting up to 4 errors on an one-case-by-one-case basis, in which the calculation of unknown syndromes is eliminated and the conditions for checking for the various numbers of errors that exist in the received word are also simplified compared to those from Lin T. C. et al.’s algorithm, which is a traditional algebraic decoding algorithm (ADA). The computational complexity of the decoder performing the binary (41, 21, 9) QR code is analyzed thoroughly, which demonstrates that the proposed decoding scheme is faster, simpler, and more suitable for implementation than Lin T. C. et al.’s algorithm. Numerical emulation results demonstrate that the proposed decoding scheme achieves the same error-rate performance as Lin T. C. et al.’s algorithm, but it has a significantly decreased the complexity of this decoder in terms of the CPU time.

Fast decoding of binary quadratic residue codes

In a recent study, Li et al. proposed an algorithm for the decoding of binary quadratic residue codes with tiny memory requirements. In this Letter, this algorithm is modified in order to dramatically improve the decoding speed. The case of the ( 89 , 45 , 17 ) binary quadratic residue code is used to illustrate the new algorithm.

Using the Difference of Syndromes to Decode Quadratic Residue Codes

In this paper, an efficient decoding algorithm is developed to facilitate faster decoding of the binary systematic quadratic residue (QR) codes. It is based on the difference of syndromes (DS), and hence, is called the DS algorithm hereinafter. This new method combines the advantages of the syndrome-weight algorithm and properties of the cyclic codes. Actually, it is a natural generalization of the cyclic weight (CW) algorithm for the (47, 24, 11) QR code developed by Lin et al., and its validity of decoding any binary systematic QR code is also proved. The complexity analysis and simulation results show that the DS algorithm dramatically reduces the decoding complexity without performance loss and considerably requires less memory when compared with previous ones. Utilizing the (47, 24, 11), the (71, 36, 11), the (73, 37, 13), and the (89, 45, 17) QR codes as examples, the DS algorithm not only significantly improves the decoding efficiency, but also saves the memory evidently. When the (23, 12, 7) QR code is considered, the DS algorithm performs almost as well as the currently known best algorithm in terms of decoding efficiency and memory requirements. Thus, all QR codes of lengths less than 100 can be decoded efficiently by using the proposed algorithm. Especially, when the proposed algorithm is applied to decode the (89, 45, 17) QR code, the best one among all the QR codes of lengths less than 100, the decoding speed is raised 26 times and the memory is saved up to 76.6% in comparison with the existing fastest decoding algorithm.

A construction of group divisible designs with block sizes 3 to 7

This paper gives a construction of group divisible designs (GDDs) on the binary extension fields with block sizes 3, 4, 5, 6, and 7, respectively, which consist of the error patterns whose first syndromes are zeros recognized from the decoding of binary quadratic residue codes. A conjecture is proposed for this construction of GDDs with larger block sizes.

New Efficient Scheme Based on Reduction of the Dimension in the Multiple Impulse Method to Find the Minimum Distance of Linear Codes

In order to find a minimum weight codeword in a linear code, the Multiple Impulse Method uses the Ordered Statistics Decoder of order 3 having a complexity which increases with the code dimension. This paper presents an important improvement of this method by finding a sub code of C of small dimension containing a lowest weight codeword. In the case of Binary Extended Quadratic Residue codes, the proposed technique consists on finding a self invertible permutation ( from the projective special linear group and searching a codeword having the minimum weight in the sub code fixed by (. The proposed technique gives the exact value of the minimum distance for all binary quadratic residue codes of length less than 223 by using the Multiple Impulse Method on the sub codes in less than one second. For lengths more than 223, the obtained results prove the height capacity of the proposed technique to find the lowest weight in less time. The proposed idea is generalized for BCH codes and it has permits to find the true value of the minimum distance for some codes of lengths 1023 and 2047. The proposed methods performed very well in comparison to previously known results.

Algebraic decoding of the binary systematic (71, 36, 11) quadratic residue code

A fast and efficient algebraic decoding algorithm (ADA) is proposed to correct up to five possible error patterns in the binary systematic (71, 36, 11) quadratic residue (QR) code. The technique required here is based on the ADA developed by He et al. [2001. Decoding the (47, 24, 11) quadratic residue code. IEEE transactions on information theory, 47 (3), 1181–1186.] and the modification of the ADA developed by Lin et al. [2010. Decoding of the (31, 16, 7) quadratic residue code. Journal of the Chinese institute of engineers, 33 (4), 573–580; 2010. High speed decoding of the binary (47, 24, 11) quadratic residue code. Information sciences, 180 (20), 4060–4068]. The new proposed conditions and the error-locator polynomials for different numbers of errors in the received word are derived. Simulation results show that the decoding speed of the proposed ADA is faster than the other existing ADAs.

Algebraic decoding of the (73, 37, 13) quadratic residue code

In this study, an efficient and fast algebraic decoding algorithm (ADA) for the binary systematic quadratic residue (QR) code of length 73 with the reducible generator polynomial to correct up to six errors is proposed. The S(I, J) matrix method given by He et al. (2001) is utilised to compute the unknown syndromes S5. A technique called swap base is proposed to correct the weight-4 error patterns. To correct the weight-5 error patterns, the new error-locator polynomials for decoding the five errors are derived. Finally, the modified shift-search algorithm (SSA) developed by Lin et al. (2010) is applied to correct the weight-6 error patterns. Moreover, the computations of all syndromes are achieved in a small finite field. Simulation results show that the proposed ADA is practical.

Fast Algebraic Decoding of the (89, 45, 17) Quadratic Residue Code

In this letter, the algebraic decoding algorithm of the (89, 45, 17) binary quadratic residue (QR) code proposed by Truong et al. is modified by using the efficient determination algorithm of the primary unknown syndromes. The correctness of the proposed decoding algorithm is verified by computer simulations and the use of two corollaries. Also, simulation results show that the CPU time of this algorithm is approximately 4 times faster than that of the previously mentioned decoding algorithm at least. Therefore, such a fast decoding algorithm can now be applied to achieve efficiently the reliability-based decoding for the (89, 45, 17) QR code. Finally, the performance of its algebraic soft-decision decoder expressed in terms of the bit-error probability versus E <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">b</sub> /N <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">0</sub> is given but not available in the literature.

Algebraic decoding of the ternary (37, 18, 11) quadratic residue code

The algebraic decoding of binary quadratic residue codes can be performed using the Peterson or the Berlekamp–Massey algorithm once certain unknown syndromes are determined. In this work, the technique of determining unknown syndromes is extended to the ternary case to decode the expurgated ternary quadratic residue code of length 37.

High speed decoding of the binary (47, 24, 11) quadratic residue code

An efficient table lookup decoding algorithm (TLDA) is presented to decode up to five possible errors in a binary systematic (47, 24, 11) quadratic residue (QR) code. The main idea of the TLDA is based on the weight of syndrome, the syndrome decoder together with a reduced-size lookup table (RSLT), and the shift-search method given by Reed et al. Thus, the size of the lookup table and computational complexity in a finite field can be significantly reduced. The memory size of the proposed condensed lookup table (CLT) consists of only 36.6 Kbytes and is only about 0.24% of the full lookup table (FLT) and 3.2% of the lookup up table given by Chen et al., respectively. These facts lead to significant reduction of computational time and the decoding complexity. A simulation result shows that the decoding speed of the proposed TLDA is much faster than all existing decoding algorithms. Moreover, it can be extended to decode all QR codes, including the class of the cyclic codes when the code length is moderate. The CLT makes this new decoding algorithm suitable for hardware or firmware implementations.

Decoding of the (31, 16, 7) quadratic residue code

An algebraic decoding algorithm is proposed to correct all error patterns of up to three errors in the binary (31, 16, 7) Quadratic Residue (QR) code with reducible generator polynomial. The decoding technique, a modification of the decoding algorithm given by Reed et al., is based on the application of the decoding algorithm proposed by Truong et al. The computation of all syndromes is done in a small field, namely, GF(25). Thus, the computational complexity can be reduced. A full simulation shows that this novel decoding method is superior to the algebraic decoding algorithm given by Reed et al.

Decoding Binary Quadratic Residue Codes Using the Euclidean Algorithm

A simplified algorithm for decoding binary quadratic residue (QR) codes is developed in this paper. The key idea is to use the efficient Euclidean algorithm to determine the greatest common divisor of two specific polynomials which can be shown to be the error-locator polynomial. This proposed technique differs from the previous schemes developed for QR codes. It is especially simple due to the well-developed Euclidean algorithm. In this paper, an example using the proposed algorithm to decode the (41, 21, 9) quadratic residue code is given and a C++ program of the proposed algorithm has been executed successfully to run all correctable error patterns. The simulations of this new algorithm compared with the Berlekamp-Massey (BM) algorithm for the (71, 36, 11) and (79, 40, 15) quadratic residue codes are shown.

Algebraic decoding of the (41, 21, 9) Quadratic Residue code

In this paper, an algebraic decoding algorithm is proposed to correct all patterns of four or fewer errors in the binary (41, 21, 9) Quadratic Residue (QR) code. The technique needed here to decode the (41, 21, 9) QR code is different from the algorithms developed in [I.S. Reed, T.K. Truong, X. Chen, X. Yin, The algebraic decoding of the (41, 21, 9) Quadratic Residue code, IEEE Transactions on Information Theory 38 (1992 ) 974–986]. This proposed algorithm does not require to solve certain quadratic, cubic, and quartic equations and does not need to use any memory to store the five large tables of the fundamental parameters in GF(2 20) to decode this QR code. By the modification of the technique developed in [R. He, I.S. Reed, T.K. Truong, X. Chen, Decoding the (47, 24, 11) Quadratic Residue code, IEEE Transactions on Information Theory 47 (2001) 1181–1186], one can express the unknown syndromes as functions of the known syndromes. With the appearance of known syndromes, one can solve Newton’s identities to obtain the coefficients of the error-locator polynomials. Besides, the conditions for different number of errors of the received words will be derived. Computer simulations show that the proposed decoding algorithm requires about 22% less execution time than the syndrome decoding algorithm. Therefore, this proposed decoding scheme developed here is more efficient to implement and can shorten the decoding time.

A New Scheme to Determine the Weight Distributions of Binary Extended Quadratic Residue Codes

This letter proposes a novel scheme which consists of a weight-counting algorithm, the combinatorial designs of the Assmus-Mattson theorem, and the weight polynomial of Gleason's theorem to determine the weight distributions of binary extended quadratic residue codes. As a consequence, the weight distributions of binary (138, 69, 22) and (168, 84, 24) extended quadratic residue codes are given.

A unified method for determining the weight enumerators of binary extended quadratic residue codes

This letter proposes an improved and unified method to determine the weight enumerators of binary extended quadratic residue (EQR) codes. It is faster than the previous methods for some of binary EQR codes. Moreover, all the results for the weight enumerators of binary EQR codes are listed.

Decoding the Ternary (23, 11, 9) Quadratic Residue Code

The algebraic decoding of binary quadratic residue codes can be performed using the Peterson or the Berlekamp‐Massey algorithm once certain unknown syndromes are determined or eliminated. The technique of determining unknown syndromes is applied to the nonbinary case to decode the expurgated ternary quadratic residue code of length 23.