On formulas for decoding binary cyclic codes

TL;DR

This paper introduces an automated algebraic decoding method for cyclic codes, including quadratic residue codes, using Gr"obner bases to derive formulas for the error locator polynomial from syndromes.

Contribution

It develops a novel approach employing elimination theory and Gr"obner bases to automatically compute decoding formulas for cyclic codes.

Findings

Automated derivation of decoding formulas for quadratic residue codes.

Use of Gr"obner bases simplifies and generalizes algebraic decoding.

Potential for improved decoding efficiency and accuracy.

Abstract

We adress the problem of the algebraic decoding of any cyclic code up to the true minimum distance. For this, we use the classical formulation of the problem, which is to find the error locator polynomial in terms of the syndroms of the received word. This is usually done with the Berlekamp-Massey algorithm in the case of BCH codes and related codes, but for the general case, there is no generic algorithm to decode cyclic codes. Even in the case of the quadratic residue codes, which are good codes with a very strong algebraic structure, there is no available general decoding algorithm. For this particular case of quadratic residue codes, several authors have worked out, by hand, formulas for the coefficients of the locator polynomial in terms of the syndroms, using the Newton identities. This work has to be done for each particular quadratic residue code, and is more and more difficult as the length is growing. Furthermore, it is error-prone. We propose to automate these computations, using elimination theory and Grbner bases. We prove that, by computing appropriate Grbner bases, one automatically recovers formulas for the coefficients of the locator polynomial, in terms of the syndroms.