环$\mathbb{F}_{p}[u]/\langle u^m-u\rangle$上的二次剩余码及其Gray象
Quadratic residue codes over the ring $\mathbb{F}_{p}[u]/\langle u^m-u\rangle$ and their Gray images
摘要：
设$m\geq 2$为任意自然数，$\mathcal{R}=\mathbb{F}_{p}+u\mathbb{F}_{p}+u^2\mathbb{F}_{p}+\cdots+u^{m-1}\mathbb{F}_{p}$为有限非链环，其中$u^m=u$和$p$是$1$模$(m-1)$的素同余.本文研究了环$\mathcal{R}$上的二次剩余码及其推广.定义了一个从$\mathcal{R}$到$\mathbb{F}_{p}^m$的灰度映射，该映射保持了线性码的自对偶性.由此构造了自对偶码、形式自对偶码和自正交码。为了说明这几个例子的自对偶，自正交和正式的自对偶码。其中构造了$\mathbb{F}_{7}$上的一个[9，3，6]线性码，该码是自正交的，并且几乎是MDS码.具有这些参数的最著名的线性码（参考文献Magma）不是自正交的。
英文摘要：
Let $m\geq 2$ be any natural number and let $\mathcal{R}=\mathbb{F}_{p}+u\mathbb{F}_{p}+u^2\mathbb{F}_{p}+\cdots+u^{m-1}\mathbb{F}_{p}$ be a finite non-chain ring, where $u^m=u$ and $p$ is a prime congruent to $1$ modulo $(m-1)$. In this paper we study quadratic residue codes over the ring $\mathcal{R}$ and their extensions. A gray map from $\mathcal{R}$ to $\mathbb{F}_{p}^m$ is defined which preserves self duality of linear codes. As a consequence self dual, formally self dual and self orthogonal codes are constructed. To illustrate this several examples of self-dual, self orthogonal and formally self-dual codes are given. Among others a [9,3,6] linear code over $\mathbb{F}_{7}$ is constructed which is self-orthogonal as well as nearly MDS. The best known linear code with these parameters (ref. Magma) is not self orthogonal.
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