On the enumeration of self-dual codes

This study presents a novel methodology for deriving both the received polynomial and the error locator polynomial for the Quadratic Residue (QR) code (57,29,17). Utilizing a newly developed algorithm, the weight distribution of this QR code is efficiently computed through Magma computations. Furthermore, the study introduces an innovative algorithm specifically de- signed to calculate Lee weights for the QR code, enhancing the accuracy and efficiency of these computations. The proposed techniques significantly contribute to the field of coding theory by improving the decoding process and providing deeper insights into the properties of QR codes. Objectives: The paper presents a novel computational framework for analyzing the Quadratic Residue (QR) code (57,29,17). To compute the weight distribution of the QR code using an efficient algorithm implemented in Magma. To derive the Lee weight distribution of the QR code using a newly developed algorithm implemented in SageMath. To construct and analyze the error locator polynomial, which aids in error detection and correction for the QR code. Methods: Constructed the generator matrix using quadratic residues modulo 57. Generated codewords by taking linear combinations of rows in the matrix. Computed the Hamming weight of each codeword. Counted occurrences of different weights to form the weight distribution table. Defined the Lee weight metric for elements in ℤ57​. Randomly generated codewords and calculated their Lee weights using parallel computing. Aggregated results to derive the Lee weight distribution of the QR code. Considered cases of 1 to 8 errors in the received polynomial. Derived the error locator polynomial for each case. Demonstrated the process of recovering the transmitted message using syndromes. Results: Successfully computed and tabulated the weight distribution of (57,29,17). Showed how different weights contribute to the structure and error-correcting capability of the code. Derived the Lee weight distribution, providing a new perspective for analyzing the QR code. Demonstrated that the Lee weight metric can be effectively applied to QR codes over finite rings. Derived error locator polynomials for different error cases. Confirmed that the QR code (57,29,17) can detect and correct up to 8 errors. Conclusions: In this study focuses on the in-depth analysis of the Quadratic Residue code (57, 29, 17) by deriving both its weight distribution and Lee weight distribution. The newly developed algorithms for calculating these distributions contribute significantly to understanding the code’s structure and its error detection and correction capabilities. Additionally, the error locator polynomial for the code has been successfully found, which further enhances its practical application in error correction

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.

The Theory of Error-Correcting 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.

In this paper, a method to search the set of syndromes' indices needed in computing the unknown syndromes for the (73, 37, 13) quadratic residue (QR) code is proposed. According to the resulting index sets, one computes the unknown syndromes and thus finds the corresponding error-locator polynomial by using an inverse-free Berlekamp-Massey (BM) algorithm. Based on the modified Chase-II algorithm, the performance of soft-decision decoding for the (73, 37, 13) QR code is given. This result is new. Moreover, the error-rate performance of linear programming (LP) decoding for the (73, 37, 13) QR code is also investigated, and LP-based decoding is shown to be significantly superior in performance to the algebraic soft-decision decoding while requiring almost the same computational complexity. In fact, the algebraic hard-decision and soft-decision decoding of the (89, 45, 17) QR code outperforms that of the (73, 37, 13) QR code because the former has a larger minimal distance. However, experimental results indicate that the (73, 37, 13) QR code outperforms the (89, 45, 17) QR code with much fewer arithmetic operations when using the LP-based decoding algorithms. The pseudocodewords analysis partially explains this seemingly strange phenomenon.

On the classification and enumeration of self-dual codes

Parametrization of self-dual codes by orthogonal matrices

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.

A method for decoding of the quadratic residue (QR) code with hash tables is presented. The method can be applied in decoding the (31, 16, 7) QR code that the generator polynomial can be factored. In other words, the mapping between elements of syndrome S1 and all correctable error patterns is not one-to-one. Therefore, the decoding of the (31, 16, 7) QR code needs to stuck S1,S5, S7 known syndrome mapping an error pattern that has one-to-one nature, where a subscript 1, 5, 7 are cyclotomic cosets. Furthermore, the algorithm determines the error locations by hash tables without operating the additions and multiplications over finite field. To decode QR code result, the hash table method for the (31, 16, 7) QR code is dramatically reduced the memory size above 97%. It is very suitable for high speed in modern communication system.

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 (41, 21, 9) Quadratic Residue code

In this paper, we study quadratic residue (QR) codes of prime length [Formula: see text] over the ring [Formula: see text] with [Formula: see text] and [Formula: see text], where [Formula: see text] and [Formula: see text] are distinct odd prime numbers. We analyze some basic properties of cyclic codes of length [Formula: see text] over [Formula: see text], we define QR codes by their generating idempotents. Further, we discuss the extended QR codes. We present a considerable number of good [Formula: see text]-ary codes as Gray images of QR codes over [Formula: see text] by considering the case when [Formula: see text] is an odd prime.

In this paper 1 , a method to search the subsets I and J needed in computing the unknown syndromes for the (73, 37, 13) quadratic residue (QR) code is proposed. According to the resulting I and J, one computes the unknown syndromes, and thus finds the corresponding error-locator polynomial by using an inverse-free BM algorithm. Based on the modified Chase-II algorithm, the performance of soft-decision decoding for the (73, 37, 13) QR code is given. This result is never seen in the literature, to our knowledge. Moreover, the error-rate performance of linear programming (LP) decoding for the (73, 37, 13) QR code is also investigated, and LP-based decoding is shown to be significantly superior in performance to the algebraic soft-decision decoding while requiring almost the same computational complexity.

In recent years, the decoding of quadratic residue (QR) codes has attracted a wide attention, considering their good properties in terms of minimal distance and special mathematic structure that make them one of the most known subclasses of codes in the family of cyclic codes. These codes are known for their complicated decoding procedure and the difficult hardware implementation. In this paper, we propose a new decoding method to decode the (17,9,5) quadratic residue code. This method does require neither the computation of the unknown syndromes nor the error-locator polynomial, instead, it decodes the discussed code by using a simple way to locate the position of the errors. To ensure the validity of the proposed method, we tested all the possible error patterns. and the results were very satisfying since the proposed decoder corrected all of them. So, we declare that this decoder can surely decode up to the to correcting capacity of this code.

We study self-dual codes (SD) over GF(2) with 8 cycles of length 11 in the decomposition of its automorphism and minimum weight at least 16. The main method we use is by Huffman and Yorgov and is for constructing SD codes having an automorphism of order p for a prime p > 2. We prove that [8],[4] SD codes over F210 such that their preimage is a binary code with minimum weight 20 does not exist. By restricting the matrices of the subcode to have symmetric or circulant submatrix in their systematic form, we have found many new Hermitian self-dual codes over F210. Those newly found Hermitian codes allow us to constructed 468062 new binary SD [88,44,16] doubly-even codes.