多项式环有限商上码的对偶
Dual of Codes over Finite Quotients of Polynomial Rings
摘要：
设$A=\frac{\mathbb{F}[x]}{\langle f(x)\rangle }$，其中$f(x)$是有限域$\mathbb{F}$上的一元多项式.本文研究了$A$ -码与其编码之间的关系。特别地，我们陈述了对Berger和El Amrani的定理的反例和修正（Codes over finite contricents of polynomial rings，\emph{Finite Fields Appl.} \textbf{25}（2014），165- 181），并提出了一种有效的算法来找到给定$A$ -码的对偶的生成元系统。我们还刻画了长度为2的自对偶$A$ -码，并研究了$A$ -码的$\mathbb{F}$ -对偶何时是$A$ -码.
英文摘要：
Let $A=\frac{\mathbb{F}[x]}{\langle f(x)\rangle }$, where $f(x)$ is a monic polynomial over a finite field $\mathbb{F}$. In this paper, we study the relation between $A$-codes and their duals. In particular, we state a counterexample and a correction to a theorem of Berger and El Amrani (Codes over finite quotients of polynomial rings, \emph{Finite Fields Appl.} \textbf{25} (2014), 165--181) and present an efficient algorithm to find a system of generators for the dual of a given $A$-code. Also we characterize self-dual $A$-codes of length 2 and investigate when the $\mathbb{F}$-dual of $A$-codes are $A$-codes.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
