On algebraic structure of polycyclic codes

Abstract

Let $ \mathbb{F}_{_q}$ be  a finite field with $q$ elements  and  $C\subseteq\mathbb{F}_{_q}^{^n}$ be a linear code of length   $n$ over $ \mathbb{F}_{_q}.$   In this paper,  we are interested  on the study of right polycyclic codes  as invariant subspace of $ \mathbb{F}_{_q}^{^n} $ by  a fixed  operator $T_{_R}.$ This approach has helped in one hand    to connect them  with   ideals  of a polynomial ring $\mathbb{F}_{_q}[x]/\langle f(X)\rangle ,$ where $f(x) $  is the minimal polynomial of $T_{_R}.$  In the other hand, it   allows to prove that the dual of a right polycyclic code is invariant by the adjoint operator of $T_{_R},$ hence  when $T_{_R} $ is  normal we prove that  the dual code of right polycyclic code  is also right polycyclic code, however when $T_{_R} $ isn't normal  the dual code   is equivalent to a right polycyclic code. Finally,  as in  cyclic case the   BCH-like and   Hartmann-Tzeng-like  bound for right polycyclic codes on Hamming distance are showed.