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. 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 \textit{$\ell$-trinomial codes}, after which we generalize our results to general polycyclic codes. We introduce an equivalence relation called \textit{$n$-equivalence}, which extends the known notion of $n$-equivalence for constacyclic codes \cite{Chen2014}. We compute the number of $n$-equivalence classes %, $ N_{(n,\ell)}$, for this relation and provide conditions under which two families of polycyclic (or $\ell$-trinomial) codes are equivalent. In particular, we prove that when $\gcd(n, n-\ell) = 1$, any $\ell$-trinomial code family is equivalent to a trinomial code family associated with the polynomial $x^n - x^{\ell} - 1$. Finally, we focus on $p^{\ell}$-trinomial codes of length $p^{\ell+r}$, where $p$ is the characteristic of $\mathbb{F}_q$ and $r$ an integer, and provide some examples as an application of the theory developed in this paper.

翻译：我们研究了与形式为 $x^n - a_{n-1}x^{n-1} - \ldots - a_1x - a_0$ 的多项式相关的多循环码族的等价性。首先考虑与三项式 $x^n - a_{\ell} x^{\ell} - a_0$（其中 $0<\ell<n$）相关的多循环码的特殊情形，我们将其称为 \textit{$\ell$-三项式码}，随后将结果推广至一般多循环码。我们引入一种称为 \textit{$n$-等价}的等价关系，该关系推广了常循环码中已知的 $n$-等价概念 \cite{Chen2014}。我们计算了该关系下的 $n$-等价类数量 $N_{(n,\ell)}$，并给出了两个多循环（或 $\ell$-三项式）码族等价的充分条件。特别地，我们证明当 $\gcd(n, n-\ell) = 1$ 时，任意 $\ell$-三项式码族均等价于与多项式 $x^n - x^{\ell} - 1$ 相关的三项式码族。最后，我们重点关注长度为 $p^{\ell+r}$ 的 $p^{\ell}$-三项式码，其中 $p$ 是 $\mathbb{F}_q$ 的特征且 $r$ 为整数，并通过实例展示本文理论的应用。