Search

It is well-known that the decoding of algebraic codes with error locators have been extensively conceived in the literature over a half century. The radical locator, which is an $\omega $ th root of error locator, has recently been discovered. This paper focuses on the two classes of radical locators. The first is complete radical locators, where all the locators are assigned to radical locators. The second is partial radical locators in that there are only a few locators being radical locators. In the former case based on complete radical locators, a new square matrix whose determinant is a univariate radical-locator polynomial is proposed. In particular, this matrix is modified to allow a large square matrix. It can be transformed by Gaussian elimination to the matrix in a row-echelon form in which zeros appearing in the diagonal entries of the current matrix are able to determine errors implicitly. Furthermore, this work further extends the previous results from the univariate case to multivariate cases. The matrix methods found herein enable one to decode a large class of cyclic codes efficiently. In the latter case, the cyclotomic cosets are a practical approach to select a small subset of positive integers. They are employed to develop partial radical locators. Finally, in contrast with the complete radical locators, the algebraic decoding methods make a natural use of partial radical locators more widely and flexibly to arbitrary cyclic codes. [ABSTRACT FROM AUTHOR]

Partial syndrome matrices and weak-locator polynomials have received considerable attention in recent years due to their applications in decoding cyclic codes. The technical contribution of this paper is threefold: 1) cyclic codes can be decoded by the newly proposed partial syndrome matrices, which generalize the previously known results on the determination of error positions; 2) a new type of polynomial associated with error locations, called radical-locator polynomial, is defined, which includes the weak-locator polynomial as a special case; and 3) a novel class of matrices with many zero entries, called row-echelon partial syndrome matrices, is presented, based on the Newton identities, for efficiently decoding cyclic codes. It is also shown that the radical-locator polynomials can be obtained from the determinants of the above-mentioned (row-echelon) partial syndrome matrices. [ABSTRACT FROM AUTHOR]

A class of cyclic codes has recently been decoded by using the weak-locator polynomials instead of the conventional error-locator polynomials. In this paper, a generalization of weak-locator polynomials to the multivariate cases, called the multivariate weak-locator polynomials, is defined, and a new matrix whose determinant can be expressed as a multivariate weak-locator polynomial is developed for cyclic codes. Moreover, a modified matrix along with Gaussian elimination found herein enables one to determine error positions precisely. The presented matrices result in a reduction of the number of syndromes when compared with the previously known matrices. The simulation shows that, for example, with the capability of correcting more errors, a considerably sophistical scheme of decoding the (97, 49, 15) binary cyclic code based on the proposed matrices is around 115 times faster than other existing decoders. Therefore, in general, it is developed to facilitate faster decoding of a large class of cyclic codes and is naturally suitable for the software implementation. [ABSTRACT FROM AUTHOR]

Cyclic codes have been widely used in many applications of communication systems and data storage systems. This paper proposes a new procedure for decoding cyclic codes up to actual minimum distance. The decoding procedure consists of two steps: 1) computation of known syndromes and 2) computation of error positions and error values simultaneously. To do so, a matrix whose all entries are syndromes is called syndrome matrix. A matrix whose entries are either syndromes or the elements of a finite field is said to be partial syndrome matrix. In this paper, two novel methods are presented to determine error positions and error values simultaneously and directly. The first method uses a new partial syndrome matrix along with Gaussian elimination. The partial syndrome matrices for binary (respectively, ternary) cyclic codes of lengths from 69 to 99 (respectively, 16 to 37) are tabulated. For some cyclic codes, the partial syndrome matrices contain unknown syndromes; the second method constructs a matrix from a system of equations, which is generated by the determinants of different partial syndrome matrices and makes use of Gaussian elimination to determine its row rank. Many more cyclic codes beyond the Bose–Chaudhuri–Hocquenghem bound can be decoded with these methods. [ABSTRACT FROM AUTHOR]

The algebraic decoding of a $p$ -ary cyclic code consists of four steps: 1) computation of the known syndromes using the received word; 2) computation of the unknown syndromes from the known syndromes; 3) computation of the error positions by a use of the Berlekamp-Massey (BM) algorithm and Chien’s search; and 4) computation of the error values by solving a linear system. This paper addresses two problems of determining the error positions and computing the unknown syndromes. To solve the first problem, a new matrix, together with Gaussian elimination instead of the BM algorithm and Chien’s search, is proposed. In this new simplified decoder, finding an error-locator polynomial is completely avoided. A main advantage of the presented decoding method is when the Bose-Chaudhuri–Hocquenghem bound is unequal to the minimum distance of the code. Some cyclic codes, which do not have $2t$ consecutive known syndromes, can be decoded up to their actual minimum distance by using the presented matrix once. To solve the second problem, two algorithms for different square matrices reported recently by Lee et al. are also provided to calculate the value of an unknown syndrome. Finally, an algebraic decoding of the $(47,23,15)$ ternary cyclic code is given. [ABSTRACT FROM AUTHOR]

The family of Zetterberg codes with parameters (2^u+1, 2^u+1-2u) for even u is one of the best known double error correcting codes because of their large code rate and high decoding speed. In this letter, we prove that when u is odd, Zetterberg codes can correct all errors of weight at most two with only 2^u+1 exceptions. Moreover, by multiplying (x-1) to the generator polynomials of Zetterberg codes with u odd, the cyclic codes generated are two-error correctable. A decoding algorithm is developed for the new family of Zetterberg codes. [ABSTRACT FROM PUBLISHER]