Algebraic decoding of negacyclic codes over ${\mathbb Z_4}$

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: our codes have minimum Lee distance at least 2t + 1, where the generator polynomial of the code has roots α, α ³, . . . , α ^2t-1 for a primitive 2nth root α of unity in a Galois extension of ${\mathbb {Z}_4}$ ; 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 ≤ t. Our treatment uses Gröbner bases, the decoding complexity is quadratic in t.