有限域上多循环码族的等价性
Equivalence of Families of Polycyclic Codes over Finite Fields
AI总结：
研究有限域上形如 $x^n - a_{n-1}x^{n-1} - \\\ldots - a_1x - a_0$ 的多项式对应的多循环码族的等价性，引入 $n$-等价关系并计算等价类数量，证明在特定条件下 $\\ell$-三项式码族等价于 $x^n - x^{\\\ell} - 1$ 对应的码族。
摘要：
我们研究有限域上形如 $x^n - a_{n-1}x^{n-1} - \\\ldots - a_1x - a_0$ 的多项式对应的多循环码族的等价性。我们从三项式 $x^n - a_{\\\ell} x^{\\\ell} - a_0$（其中 $0<\\\ell<n$）对应的多循环码的特例开始，我们称之为 $\\\ell$-三项式码，之后将结果推广到一般多循环码。我们引入一种称为 $n$-等价的等价关系，它扩展了常循环码的已知 $n$-等价概念 \\\cite{Chen2014}。我们计算了该关系下的 $n$-等价类的数量，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，任何 $\\\ell$-三项式码族等价于多项式 $x^n - x^{\\\ell} - 1$ 对应的三项式码族。最后，我们关注长度为 $p^{\\\ell+r}$ 的 $p^{\\\ell}$-三项式码，其中 $p$ 是 $\\\mathbb{F}_q$ 的特征，$r$ 是整数，并给出一些例子作为本文所发展理论的应用。
英文摘要：
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.
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