Algebraic Decoding of Negacyclic Codes Over Z_4

TL;DR

This paper explores algebraic decoding of negacyclic codes over Z_4, showing they can correct errors up to a certain Lee weight without restrictions on code parameters, using Grobner bases for efficient decoding.

Contribution

It introduces a new algebraic decoding algorithm for negacyclic codes over Z_4 that corrects errors up to Lee weight t, employing Grobner bases for improved efficiency.

Findings

Codes have minimum Lee distance at least 2t+1.

Decoding algorithm corrects errors of Lee weight up to t.

Decoding complexity is quadratic in t.

Abstract

In this article we investigate Berlekamp's negacyclic codes and discover that these codes, when considered over the integers modulo 4, do not suffer any of the restrictions on the minimum distance observed in Berlekamp's original papers. The codes considered here have minimim Lee distance at least 2t+1, where the generator polynomial of the code has roots z,z^3,...,z^{2t+1} for a primitive 2nth root of unity z in a Galois extension of Z4. No restriction on t is imposed. We present an algebraic decoding algorithm for this class of codes that corrects any error pattern of Lee weight at most t. Our treatment uses Grobner bases and the decoding complexity is quadratic in t.