Equivalence of Classes of Polycyclic and Skew Polycyclic Codes over Finite Fields
Vortrag von Dr. Hassan Ou-azzou
Datum: 19.02.25 Zeit: 15.15 - 16.15 Raum: Y27H28
We study the equivalence of families of polycyclic codes associated with polynomials of the form \(x^n - a_{n-1}x^{n-1} - \ldots - a_1x - a_0\) over a finite field, \(\mathbb{F}_q,\ q=p^s\). We begin with the specific case of polycyclic codes associated with a trinomial \(x^n - a_{\ell} x^{\ell} - a_0\) (for some \(0< \ell <n\)), which we refer to as \(\ell\)-trinomial codes, after which we generalize our results to general polycyclic codes. We introduce an equivalence relation called \(n\)-equivalence, which extends the known notion of \(n\)-equivalence for constacyclic codes. We compute the number of \(n\)-equivalence classes for this relation and provide conditions under which two families of polycyclic (or \(\ell\)-trinomial) codes are equivalent. In particular, we prove that when \(\gcd(n, n-\ell, q-1) = 1\), any \(\ell\)-trinomial code family is equivalent to a trinomial code family associated with the polynomial \(x^n - x^{\ell} - 1\). For \(n=p^r\), we give a complete classification of \(p^{r-1}\)-trinomial codes, and \(p^{r-s}\)-trinomial codes. Secondly, we extend our study to the case of skew polycyclic codes. Finally, we provide some examples as an application of the theory developed in this paper.