Equivalence of Polycyclic and Skew Polycyclic Codes (Part 2)
Vortrag von Dr. Hassan Ou-azzou
Datum: 19.03.25 Zeit: 15.15 - 16.15 Raum: Y27H28
In this presentation, we study the equivalence of families of skew polycyclic codes associated with the skew polynomial \( x^n - \vec{a}(x)\), with \(\vec{a}=(a_0,a_1,\ldots,a_{n-1}) \in \mathbb{F}_q^n,\) for short we say skew \((\vec{a},\sigma)\)-polycyclic code, where \(\sigma\) is automorphism of \(\mathbb{F}_q.\) We begin with the case of skew \(\ell\)-trinomial codes associated with a trinomial \(x^n - a_{\ell} x^{\ell} - a_0\) (for some \(0< \ell <n\)), after which we generalize our results to general skew polycyclic codes. We introduce an equivalence relation called {\(n\)-equivalence}, which extends the known notion of \(n\)-equivalence for skew 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. Finally, we provide some examples as an application.