Abstract

In this paper we give three constructions of cyclic self-orthogonal codes over \(\mathbb {Z}_{2^k}\) , for \(k\ge 3,\) from Boolean functions on n variables. The first construction for each k, \(3\le k\le n,\) yields a self-orthogonal \(\mathbb {Z}_{2^k}\) -code of length \(2^{n+2}\) with all Euclidean weights divisible by \(2^{k+1}.\) In the remaining two constructions, for each even n and \(k\ge 3,\) we generate a self-orthogonal \(\mathbb {Z}_{2^k}\) -code of length \(2^{n+1}.\) All Euclidean weights in the constructed code are divisible by \(2^{2k-1}\) or \(2^{k+1}\) , depending on which of the two constructions is used.

中文翻译：

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自正交 $$\mathbb {Z}_{2^k}$$ 代码的构造

摘要

本文给出了 \(\mathbb {Z}_{2^k}\) 上循环自正交码的三种构造，对于 n< 上的布尔函数的 \(k\ge 3,\) a i=6> 变量。每个 k, \(3\le k\le n,\) 的第一个构造 产生自正交 \(\mathbb {Z}_{2^k}\) 长度为 \(\mathbb {Z}_{2^k}\)，具体取决于使用两种结构中的哪一种。\(2^{k+1}\ ) 或 \(2^{2k-1}\)构造的代码中的所有欧氏权重都是可整除的通过 \(2^{n+1}.\)-长度的代码\(\mathbb {Z}_{2^k}\)我们生成一个自正交 \(k\ge 3,\) 和 n 在其余两个结构中，对于每个偶数 \(2^{k+1} 整除.\) 所有欧几里得权重均可被 \(2^{n+2}\)