A Generalization of Negacyclic Codes

In this paper, for k ≥ 1 an isometry φk between codes over Z2k+1 and codes over Z4 is introduced and is used to give a generalization of the Gray map. Furthermore, by means of this isometry, the concept of negacyclic codes is extended to codes over the ring Z2k+1, referred to as hpo-cyclic codes (half plus one-cyclic codes). A characterization of these codes in terms of their images under φk is given. It is also shown that the generalized Gray map image of a hpo-cyclic code is a binary distance invariant (not necessary linear) quasi-cyclic code. Finally, some linear hpo-cyclic codes are discussed.