Negacyclic and cyclic codes over Z4

The negashift nu of Z(4)(n) is defined as the permutation of Z(4)(n) such that nu(a(0), a(1)..., a(i),..., a(n-1)) = (-a(n-1), a(0),..., a(i),..., a(n-2)) and a negacyclic code of length n over Z(4) is defined as a subset C of Z(4)(n) such that nu(C) = C. We prove that the Gray image of a linear negacyclic code over Z(4) Of length n is a binary distance invariant (not necessary linear) cyclic code. We also prove that, if n is odd, then every binary code which is the Gray image of a linear cyclic code over Z(4) Of length n is equivalent to a (not necessary linear) cyclic code and this equivalence is explicitely described. This last result explains and generalizes the existence, already known, of versions of Kerdock, Preparata, and others codes as doubly extended cyclic codes. Furthermore, we introduce a family of binary linear cyclic codes which are Gray images of Z(4)-linear negacyclic codes.