Nonlinear cyclic codes over Z_4 whose Nechaev-Gray images are binary linear cyclic codes

Abstract

It has been previously shown [5], that a binary linear cyclic code of length 2n (n odd) can be obtained from two binary linear cyclic codes of length n by the well known |u|u+v| construction. It is easy to show that the same construction can as well be obtained as the image under the Gray map of a cyclic (not necessarily linear) code of length n over ZZ4. We shall show that the set of linear cyclic codes over ZZ4, whose images under the Gray map agree with the |u|u+v| construction, are the same family of codes whose images correspond to binary linear cyclic codes of length 2n under the Nechaev-Gray map introduced in [11]. Since the number of linear cyclic codes of length n over ZZ4 is equal to the number of binary linear cyclic codes of length 2n, we use this result to characterize the set of nonlinear cyclic codes of length n over ZZ4, whose images, under the Nechaev-Gray map, are binary linear cyclic codes of length 2n. As a byproduct, we introduce a new product for binary polinomials, and, by means of this product, we obtain a new way to express codewords that belong to a linear cyclic code over ZZ4 whose Nechaev-Gray image is a binary linear cyclic codes.