Negacyclic and cyclic codes over Z4

The negashift ν of Z₄ⁿ is defined as the permutation of Z₄ⁿ such that ν(a₀, a ₁, ···, 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 Z₄ ⁿ such that ν(C)=C. We prove that the Gray image of a linear negacyclic code over Z₄ 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 Z₄ 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 Z₄ linear negacyclic codes