Some constacyclic codes over Z_(2k) and binary quasi-cyclic codes

摘要

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 ψ~k between codes over Z_(2k+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_(2k+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 ψ~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.