On the algebraic structure of polycyclic codes

Abstract

In this paper, we are interested in the study of the right polycyclic codes as invariant subspaces of Fnq by a fixed operator TR. This approach has helped in one hand to connect them to the ideals of the polynomials ring Fq [x]/?f)X)?, where f (x) is the minimal polynomial of TR. On the other hand, it allows to prove that the dual of a right polycyclic code is invariant by the adjoint operator of TR. Hence, when TR is normal we prove that the dual code of a right polycyclic code is also a right polycyclic code. However, when TR isn?t normal the dual code is equivalent to a right polycyclic code. Finally, as in the cyclic case, the BCH-like and Hartmann-Tzeng-like bounds for the right polycyclic codes on Hamming distance are derived.