$ {{\mathbb{Z}}_{2}}{{\mathbb{Z}}_{2}}{{\mathbb{Z}}_{4}}$-additive cyclic codes

Tingting Wu; Jian Gao; Yun Gao; Fang-Wei Fu

doi:10.3934/amc.2018038

Article Contents

2018, Volume 12, Issue 4: 641-657. Doi: 10.3934/amc.2018038

This issue Previous Article Characterization of extended Hamming and Golay codes as perfect codes in poset block spaces Next Article On the dual codes of skew constacyclic codes

$ {{\mathbb{Z}}_{2}}{{\mathbb{Z}}_{2}}{{\mathbb{Z}}_{4}}$-additive cyclic codes

1.
College of Science, Civil Aviation University of China, Tianjin 300300, China
2.
School of Mathematics and Statistics, Shandong University of Technology, Zibo, Shandong 255091, China
3.
Chern Institute of Mathematics and LPMC, Nankai University, Tianjin 300071, China

* Corresponding author: Jian Gao
* Corresponding author: Jian Gao

Received: March 31, 2016

Published: October 2018

This research is supported by the National Natural Science Foundation of China (Grant Nos. 11701336, 11626144, 11671235, 61571243 and 61171082), the Scientific Research Foundation of Civil Aviation University of China (Grant No. 2017QD22X)

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Abstract

This paper is concerned with ${{\mathbb{Z}}_{2}}{{\mathbb{Z}}_{2}}{{\mathbb{Z}}_{4}}$-additive cyclic codes. These codes can be identified as submodules of the ring ${\mathbb{Z}}_{2}[x]/\langle x^r-1\rangle × {\mathbb{Z}}_{2}[x]/\langle x^s-1\rangle × {\mathbb{Z}}_{4}[x]/\langle x^t-1\rangle$. There are two major ingredients. First, we determine the generator polynomials and minimum generating sets of this kind of codes. Furthermore, we investigate their dual codes. We determine the structure of the dual of separable ${{\mathbb{Z}}_{2}}{{\mathbb{Z}}_{2}}{{\mathbb{Z}}_{4}}$-additive cyclic codes completely. For the dual of non-separable ${{\mathbb{Z}}_{2}}{{\mathbb{Z}}_{2}}{{\mathbb{Z}}_{4}}$-additive cyclic codes, we give their structural properties in a few special cases.

Keywords:

$ {{\mathbb{Z}}_{2}}{{\mathbb{Z}}_{2}}{{\mathbb{Z}}_{4}}$-cyclic codes,
generator polynomials,
minimum generating sets,
dual code.

Mathematics Subject Classification: Primary: 94B05, 94B15; Secondary: 11T71.

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References

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