On Efficient Decoding of Alternant Codes over a Commutative Ring⋆

Abstract

Let R be a commutative ring e.g. the domain of p-adic integers or a Galois ring. We define alternant codes over R, which includes BCH and Reed-Solomon codes. We also define a corresponding key equation and concentrate on decoding alternant codes when R is a domain or a local ring. Our approach is based on minimal realization (MR) of a finite sequence [4,5], which is related to rational approximation and shortest linear recurrences. The resulting algorithms have quadratic complexity.When R is a domain, the error-locator polynomial is the unique monic minimal polynomial of the finite syndrome sequence (Theorem 2), and can be easily obtained using Algorithm MR of [4] (which is division-free). The error locations and magnitudes can then be computed as over a field. In this way we can efficiently decode any alternant code over a domain.Recall that a Hensel ring is a local ring which admits Hensel lifting. (It is well-known that a finite local ring, such as a Galois ring, is a Hensel ring.) We characterize the set of monic minimal polynomials of a finite syndrome sequence over a Hensel ring (Theorem 3). It turns out that the monic minimal polynomials coincide modulo the maximal ideal M of R (Theorem 4) when R is a local ring. This yields an efficient new decoding algorithm (Algorithm 1) for alternant codes over a local ring R, once a monic minimal polynomial of the syndrome sequence is known. For determining the error locations, it is enough to find the roots of the image of any such monic minimal polynomial in the residue field R/M. After determining the error locations, the error magnitudes can be easily computed.When R is a finite chain ring (e.g. a Galois ring) we invoke Algorithm MP of [5] to find a monic minimal polynomial.We note that a modification of the Berlekamp-Massey algorithm for ℤm was given in [8], where it was claimed [loc. cit., Introduction] (without proof) to decode BCH codes defined over the integers modulo m. An algorithm to decode BCH and Reed-Solomon codes over a Galois ring has also been given in [3]. However this algorithm may require some searching see [loc. cit., Conclusions, p. 1019] and their decoding algorithm requires root-finding in R itself, which is also less efficient.For more details and proofs, we refer the reader to [7].