Gray transform for negacyclic and cyclic codes over Z₄ and F₂
by
Jacques Wolfmann
Groupe d'Etude du Codage de Toulon, Universit/'e de Toulon et du Var

Z₄ is the ring of integers modulo 4 and F₂ is the finite field of order 2. The negashift \nu of Z₄ⁿ is defined as the permutation of Z₄ⁿ such that \nu(a₀, a₁, ... , a_i, ... , a_n-1) = (-a_n-1, a₀, ... , a_i, ... , a_n-2) and a negacyclic code of length n over Z₄ is defined as a subset C of Z₄ⁿ such that \nu(C) = C.

Identifying Z₄ⁿ with the factor ring Z₄[x]/(xⁿ+1), negacyclic codes of length n are the ideals of this factor rings. Recently, the Gray map of Z₄ⁿ into F₂²ⁿ was used to solve an important old problem in Coding Theory about Kerdock and Preparata codes and this gives a new point of view on some binary codes.

We prove that the Gray map image of a linear negracyclic code over Z₄ of length n is a binary distance invariant (not necessarily linear) cyclic code. We also prove that, if n is odd, then every binary code which is the Gray map image of a linear cyclic code over Z₄ of length n is equivalent to a (not necessarily linear) cyclic code and this equivalence is explicitely described. This last result explains and generalizes the existence, already known, of versions as doubly extended cyclic codes of Kerdock, Preparata, and other non-linear binary codes.

In general, the Gray image of a Z₄-linear code is neither linear nor cyclic. We introduce a family of binary linear cyclic codes which are Gray map images of Z₄-linear negacyclic codes.

Date received: February 15, 1999