Journal of Applied Mathematics and Computing ( IF 2.2 ) Pub Date : 2020-08-28 , DOI: 10.1007/s12190-020-01425-5 Raj Kumar , Maheshanand Bhaintwal
In this paper, we study \((1+2^{s-1}u)\)-constacyclic codes and a class of skew \((1+2^{s-1}u)\)-constacyclic codes of odd length over the ring \(R= {\mathbb {Z}}_{2^s}+u{\mathbb {Z}}_{2^s}\), \(u^2=0\), where \(s \ge 3\) is an odd integer. We have obtained the algebraic structure of \((1+2^{s-1}u)\)-constacyclic codes over R. Three new Gray maps from R to \({\mathbb {Z}}_2+u{\mathbb {Z}}_2\) have been defined and it is shown that Gray images of \((1+2^{s-1}u)\)-constacyclic codes and skew \((1+2^{s-1}u)\)-constacyclic codes are cyclic codes, quasi-cyclic codes or codes that are permutation equivalent to quasi-cyclic codes over \({\mathbb {Z}}_2+u{\mathbb {Z}}_2\). Using Magma, some good cyclic codes of length 6 over \({\mathbb {Z}}_2+u{\mathbb {Z}}_2\) are obtained.
中文翻译:
在$$ \ pmb {\ mathbb {Z}} _ {2 ^ s} + u \ pmb {\ mathbb {Z}} _ {2 ^ s} $$ Z 2 s +上的一类恒定周期代码和斜周期代码u Z 2 s及其灰度图像
在本文中,我们研究\((1 + 2 ^ {s-1} u)\)-常数码和一类偏斜\((1 + 2 ^ {s-1} u)\)-常数码环上的奇数长度\(R = {\ mathbb {Z}} _ {2 ^ s} + u {\ mathbb {Z}} _ {2 ^ s} \),\(u ^ 2 = 0 \),其中\(s \ ge 3 \)是一个奇数整数。我们已获得的代数结构\((1 + 2 ^ {S-1} U)\)超过-constacyclic码ř。从R到\({\ mathbb {Z}} _ 2 + u {\ mathbb {Z}} _ 2 \)的三个新Gray映射已定义,并且显示\((1 + 2 ^ {s- 1} u)\)-常数代码和偏斜\((1 + 2 ^ {s-1} u)\)-恒定循环代码是循环代码,准循环代码或与\({\ mathbb {Z}} _ 2 + u {\ mathbb {Z}} _ 2 \)上的准循环代码等效的置换代码。使用岩浆,可以获得一些\({\ mathbb {Z}} _ 2 + u {\ mathbb {Z}} _ 2 \)上长度为6的良好循环码。