环GR(4,2)上一类负循环码的Gray象

doi:10.12341/jssms08948

系统科学与数学 ›› 2010, Vol. 30 ›› Issue (3): 334-340.DOI: 10.12341/jssms08948

环GR(4,2)上一类负循环码的Gray象

开晓山, 朱士信

合肥工业大学数学学院, 合肥 230009

收稿日期:2008-11-06 修回日期:1900-01-01 出版日期:2010-03-25 发布日期:2010-03-25

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开晓山;朱士信. 环GR(4,2)上一类负循环码的Gray象[J]. 系统科学与数学, 2010, 30(3): 334-340.

KAI Xiaoshan;ZHU Shixin. Gray Images of a Class of Negacyclic Codes over GR(4,2)[J]. Journal of Systems Science and Mathematical Sciences, 2010, 30(3): 334-340.

Gray Images of a Class of Negacyclic Codes over GR(4,2)

KAI Xiaoshan, ZHU Shixin

School of Mathematics, Hefei University of Technology, Hefei 230009

Received:2008-11-06 Revised:1900-01-01 Online:2010-03-25 Published:2010-03-25

研究了$GR(4,2)$上长为$2^{s}$的负循环码的Gray象,证明了$GR(4,2)$上长为$2^{s}$的负循环码的Gray象是$F_{4}$上长为$2^{s+2}$指数为2的准循环码. 通过计算$GR(2^{a},m)$上长为$2^{s}$的负循环码的齐次距离, 确定了$GR(4,2)$上长为$2^{s}$的负循环码的Gray象的汉明距离.

关键词： Gray象, 汉明距离, 齐次距离, 负循环码, 准循环码.

This paper investigates the Gray images of negacyclic codes of length $2^{s}$ over $GR(4,2)$. It is proved that the Gray image of a negacyclic code of length $2^{s}$ over $GR(4,2)$ is a distance-invariant linear quasicyclic code of index 2 and length $2^{s+2}$ over $F_{4}$. The homogeneous distances of negacyclic codes of length $2^{s}$ over $GR(2^{a},m)$ in general are calculated. The homogeneous distances are used to determine the Hamming distances of the Gray images of negacyclic codes of length $2^{s}$ over $GR(4,2)$.

Key words: Gray map, Hamming distance, homogeneous distance, negacyclic codes, quasicyclic codes.

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