On the Shape of the General Error Locator Polynomial for Cyclic Codes

Abstract

A general result on the explicit form of the general error locator polynomial for all cyclic codes is given, along with several results for infinite classes of cyclic codes with t = 2 and t = 3. From these, a theoretically justification of the sparsity of the general error locator polynomial is obtained for all cyclic codes with t ≤ 3 and n < 63, except for three cases where the sparsity is proved by a computer check. Moreover, we discuss some consequences of our results to the understanding of the complexity of bounded-distance decoding of cyclic codes. DRAFT at most t errors. Unfortunately, for an arbitrary cyclic code the number of consecutive known syndromes is less than 2t. When few unknown syndromes are needed to get 2t consecutive syndromes, a good strategy could be to develop an efficient method to determine expressions of unknown syndromes in terms of known syndromes.In [3] Feng and Tzeng proposed a matrix method which is based on the existence of a syndrome matrix with a particular structure. This method depends on the weight of the error pattern, so it leads to a step-by-step decoding algorithm, and hence the error locator polynomial may not be determined in one step. In [4] He et al. developed a modified version of the Feng-Tzeng method, and used it to determine the needed unknown syndrome and to decode the binary QR code of length 47. In [5], [6], [7] Chang et al. presented algebraic decoders for other binary QR codes combining the Feng-He matrix method and the BM algorithm. Another method used to yield representations of unknown syndromes in terms of known syndromes is the Lagrange interpolation formula (LIF)[8]. This method has two main problems: it can be applied only to codes generated by irreducible polynomials and its computational time grows substantially as the number of errors increases.The first problem was overcome by Chang et al. in [9]. Here the authors introduced a multivariate interpolation formula (MVIF) over finite fields and used it to get an unknown syndrome representation method similar to that in [8]. They also apply the MVIF to obtain the coefficients of the general error locator polynomial of the [15, 11, 5] Reed-Solomon (RS) code. Later, trying to overcome the second problem, Lee et al.[10] presented a new algorithm which combines the syndrome matrix search and the modified Chinese remainder theorem (CRT). Compared to the Lagrange interpolation method, this substantially reduces the computational time for binary cyclic codes generated by irreducible polynomials. Besides the unknown syndrome representation method, other approaches have been proposed to decode cyclic codes. In 1987 Elia [11] proposed a seminal efficient algebraic decoding algorithm for the Golay code of length 23. Orsini and Sala [12] introduced the general error locator polynomial and presented an algebraic decoderwhich permits to determine the correctable error patterns of a cyclic code in one step. They constructively showed that general error locator polynomials exist for any cyclic code (it is shown to exis...

HELP: a sparse error locator polynomial for BCH codes

“…This approach was improved in a series of paper [5,27] culminating with [32] which, specializing Gianni-Kalkbrener Theorem [22,24], stated Theorem 6 below. For a survey of this Cooper Philosophy see [30] and on Sala-Orsini locator [6].…”