Decoding Procedure for BCH, Alternant and Goppa Codes defined over Semigroup Ring
نویسنده
چکیده
In this paper we present a decoding principle for BCH, Alternant and Goppa codes constructed through a semigroup ring, which is based on modified Berlekamp-Massey algorithm. This algorithm corrects all errors up to the Hamming weight t ≤ r, i.e., whose minimum Hamming distance is 2r + 1. Key-word: Semigroup rings, BCH codes, Alternant codes, Goppa codes, modified BerlekampMassey algorithm.
منابع مشابه
Encoding through generalized polynomial codes
This paper introduces novel constructions of cyclic codes using semigroup rings instead of polynomial rings. These constructions are applied to define and investigate the BCH, alternant, Goppa, and Srivastava codes. This makes it possible to improve several recent results due to Andrade and Palazzo [1]. Mathematical subject classification: 18B35, 94A15, 20H10.
متن کاملAlternant and BCH codes over certain rings
Alternant codes over arbitrary finite commutative local rings with identity are constructed in terms of parity-check matrices. The derivation is based on the factorization of xs − 1 over the unit group of an appropriate extension of the finite ring. An efficient decoding procedure which makes use of the modified Berlekamp-Massey algorithm to correct errors and erasures is presented. Furthermore...
متن کاملCodes through Monoid Rings and Encoding
Cazaran and Kelarev [2] have given necessary and sufficient conditions for an ideal to be the principal; further they described all finite factor rings Zm[X1, · · · , Xn]/I, where I is an ideal generated by an univariate polynomial, which are commutative principal ideal rings. But in [3], Cazaran and Kelarev characterize the certain finite commutative rings as a principal ideal rings. Though, t...
متن کاملLinear Codes over Finite Rings
Linear codes over finite rings with identity have recently raised a great interest for their new role in algebraic coding theory and for their successful application in combined coding and modulation. Thus, in this paper we address the problems of constructing of new cyclic, BCH, alternant, Goppa and Srivastava codes over local finite commutative rings with identity. These constructions are ver...
متن کاملOn the Key Equation Over a Commutative Ring
We define alternant codes over a commutative ring R and a corresponding key equation. We show that when the ring is a domain, e.g. the p-adic integers, the error–locator polynomial is the unique monic minimal polynomial (shortest linear recurrence) of the syndrome sequence and that it can be obtained by Algorithm MR of Norton. When R is a local ring, we show that the syndrome sequence may have ...
متن کامل