Some constacyclic codes over Z2 k and binary quasi-cyclic codes

Abstract

The concept of negacyclic code was recently introduced in Wolfmann (IEEE Trans. Inform. Theory 45 (1999) 2527–2532), in which some relations between the negacyclic codes and their Gray map images are proved. In this note, for k⩾1 an isometry ϕ k between codes over Z 2 k+1 and codes over Z 4 is introduced and used to give a generalization of the Gray map equivalent to the one given in Carlet (IEEE Trans. Inform. Theory 44 (1998) 1543–1547). Furthermore, by means of this isometry, the concept of negacyclic codes is extended to codes over the ring Z 2 k+1 , obtaining a class of constacyclic codes referred to as hpo-cyclic codes (half plus one-cyclic codes). A characterization of these codes in terms of their images under ϕ k is given. It is also proved that the generalized Gray map image of an hpo-cyclic code is a binary distance invariant (not necessarily linear) quasi-cyclic code. Finally, some linear hpo-cyclic codes are discussed and a few examples are given.