Finite Fields and Their Applications ( IF 1.2 ) Pub Date : 2021-09-15 , DOI: 10.1016/j.ffa.2021.101923 Bohyun Kim 1 , Yoonjin Lee 1 , Jinjoo Yoo 2
Our aim for this paper is to find the construction method for quasi-cyclic self-orthogonal codes over the finite field . We first explicitly determine the generators of α-constacyclic codes over the finite Frobenius non-chain ring , where m is a positive integer, is a unit of , , and a is nonzero. We then find a Gray map from (with respect to homogeneous weights) to (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 of length n, which produces infinitely many quasi-cyclic self-orthogonal codes over of length and index . 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.
中文翻译:
一个无限的 Griesmer 拟循环自正交码族
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 . 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,其中m是一个正整数, 是一个单位 , ,且a非零。然后我们找到一张灰色地图 (关于均质权重)到 (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码长度为n,它产生无限多个准循环自正交码 长度 和索引 . 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。