Our aim for this paper is to find the construction method for quasi-cyclic self-orthogonal codes over the finite field ${\mathbb{F}}_{p^m}$. We first explicitly determine the generators of α-constacyclic codes over the finite Frobenius non-chain ring $R_{p,m}={\mathbb{F}}_{p^m}[u,v]/〈u^2=v^2=0,uv=vu〉$, where m is a positive integer, $\alpha =a+ub+vc+uvd$ is a unit of $R_{p,m}$, $a,b,c,d\in {\mathbb{F}}_{p^m}$, and a is nonzero. We then find a Gray map from $R_{p,m}[x]/〈x^n-\alpha 〉$ (with respect to homogeneous weights) to ${\mathbb{F}}_{p^m}[x]/〈x^{p^{3m+1}n}-a〉$ (with respect to Hamming weights), which is linear and preserves minimum weights. We present an efficient algorithm for finding the Gray images of α-constacyclic codes over $R_{p,m}$ of length n, which produces infinitely many quasi-cyclic self-orthogonal codes over ${\mathbb{F}}_{p^m}$ of length $p^{3m+1}$ and index $p^{3m}$. In particular, some family turns out to be “Griesmer” codes; these Griesmer quasi-cyclic self-orthogonal codes are “new” codes compared with previously known Griesmer codes of dimension 4.

我们这篇论文的目的是找到有限域上拟循环自正交码的构造方法 ${\mathbb{F}}_{{磷}^{米}}$. 我们首先明确地确定有限 Frobenius 非链环上的α -恒环码的生成器${\mathrm{电阻}}_{磷,米}={\mathbb{F}}_{{磷}^{米}}[你,v]/〈{你}^2=v^2=0,你v=v你〉$，其中m是一个正整数，$\alpha =\mathrm{一种}+你乙+vC+你vd$ 是一个单位 ${\mathrm{电阻}}_{磷,米}$, $\mathrm{一种},乙,C,d\in {\mathbb{F}}_{{磷}^{米}}$，且a非零。然后我们找到一张灰色地图${\mathrm{电阻}}_{磷,米}[X]/〈X^n-\alpha 〉$ （关于均质权重）到 ${\mathbb{F}}_{{磷}^{米}}[X]/〈X^{{磷}^{3米+1}n}-\mathrm{一种}〉$（相对于汉明权重），它是线性的并保持最小权重。我们提出了一个高效的算法发现的灰度图像α超过-constacyclic码${\mathrm{电阻}}_{磷,米}$长度为n，它产生无限多个准循环自正交码${\mathbb{F}}_{{磷}^{米}}$ 长度 ${磷}^{3米+1}$ 和索引 ${磷}^{3米}$. 特别是，一些家庭原来是“Griesmer”代码；与先前已知的第 4 维 Griesmer 码相比，这些 Griesmer 准循环自正交码是“新”码。