On polycyclic linear and additive codes associated to a trinomial over a finite chain ring

Abstract

In this paper, we investigate polycyclic codes associated with a trinomial of arbitrary degree $n$ over a finite chain ring $R.$ We extend the concepts of $n$-isometry and $n$-equivalence known for constacyclic codes to this class of codes, providing a broader framework for their structural analysis. We describe the classes of $n$-equivalence and compute their number, significantly reducing the study of trinomial codes over $R$. Additionally, we examine the special case of trinomials of the form $x^n-a_{1}x-a_0\in R[x]$ and analyze their implications. Finally, we consider the extension of our results to certain trinomial additive codes over $R.$