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arXiv · 1501.01360

On double cyclic codes over Z_4

Jian Gao·Minjia Shi·Tingting Wu·Fang-Wei Fu
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Abstract

Let $R=\mathbb{Z}_4$ be the integer ring mod $4$. A double cyclic code of length $(r,s)$ over $R$ is a set that can be partitioned into two parts that any cyclic shift of the coordinates of both parts leaves invariant the code. These codes can be viewed as $R[x]$-submodules of $R[x]/(x^r-1)\times R[x]/(x^s-1)$. In this paper, we determine the generator polynomials of this family of codes as $R[x]$-submodules of $R[x]/(x^r-1)\times R[x]/(x^s-1)$. Further, we also give the minimal generating sets of this family of codes as $R$-submodules of $R[x]/(x^r-1)\times R[x]/(x^s-1)$. Some optimal or suboptimal nonlinear binary codes are obtained from this family of codes. Finally, we determine the relationship of generators between the double cyclic code and its dual.

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Jian Gao, Minjia Shi, Tingting Wu, Fang-Wei Fu. 2015-01-07. On double cyclic codes over Z_4. https://arxiv.org/abs/1501.01360

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