有限域上最优码的一种新的构造方法

doi:10.3969/j.issn.0372-2112.2015.08.029

电子学报 ›› 2015, Vol. 43 ›› Issue (8): 1662-1667.DOI: 10.3969/j.issn.0372-2112.2015.08.029

有限域上最优码的一种新的构造方法

丁健, 李红菊

安徽新华学院公课部, 安徽合肥 230088

收稿日期:2014-12-17 修回日期:2015-04-27 出版日期:2015-08-25
- 作者简介:
- 丁健 男,1982年生于安徽合肥.安徽新华学院讲师.研究方向为代数编码与密码. E-mail:dingjian_happy@163.com 李红菊 女,1982年生于安徽宿州.安徽新华学院讲师.研究方向为代数编码与密码.
- 基金资助:
- 安徽省教育厅自然科学项目 (No.KJ2013B107); 安徽新华学院校级科研项目、质量工程项目 (No.2014Zr009,No.2013tskcx06,No.2013jgkcx09,No.2014xgg09)

A New Method on the Construction of Optimal Codes over Finite Field

DING Jian, LI Hong-ju

Department of Common Course, Anhui Xinhua University, Hefei, Anhui 230088, China

Received:2014-12-17 Revised:2015-04-27 Online:2015-08-25 Published:2015-08-25
- Supported by:
- Natural Scientific Program of Department of Education of Anhui Province (No.KJ2013B107); Research Program and Quality Engineering Project of Anhui Xinhua University (No.2014Zr009, No.2013tskcx06, No.2013jgkcx09, No.2014xgg09)

摘要/Abstract

摘要：

基于域F_p^m上一类特殊的矩阵,定义了环R(_p^m,k)=F_p^m[u]/<u^k>到F_p^{p_m^j}的一个新的Gray映射,其中u^k=0、p为素数、j为正整数且p^j－1+1≤k≤p^j.得到了环R(_p^m,k)上码长为任意长度N的(1+u)常循环码的Gray象是F_p^m上长为p^jN的保距线性循环码,并给出了Gray象的生成多项式,构造了F₃,F₅和F₇上的一些最优线性循环码.

关键词: 线性码, 循环码, 常循环码, Gray映射, 最优码

Abstract:

Let R(_p^m, k) denote the polynomial residue ring F_p^m[u]/<u^k>, where p^j－1+1≤k≤p^j and u^k=0 for some positive prime number p and positive integer j.Based on a kind of matrix over F_p^m, a new Gray map from R(_p^m, k) to F_p^{p_m^j} is defined.It is proved that the Gray image of a linear (1+u) constacyclic code of an arbitrary length N over R(_p^m, k) is a distance invariant linear cyclic code of length p^jN over F_p^m.Moreover, the generator polynomial of the Gray image of such a constacyclic code is determined, and some optimal linear cyclic codes over F₃, F₅ and F₇ are constructed via the Gray map.

Key words: linear code, cyclic code, constacyclic code, Gray map, optimal code

中图分类号:

TN911.22

丁健, 李红菊. 有限域上最优码的一种新的构造方法[J]. 电子学报, 2015, 43(8): 1662-1667.

DING Jian, LI Hong-ju . A New Method on the Construction of Optimal Codes over Finite Field[J]. Acta Electronica Sinica, 2015, 43(8): 1662-1667.

参考文献

[1] A R Hammons,P V Kumar,A R Calderbank,N J A Sloane,P Solé.The Z₄-linearity of Kerdock,Preparata,Goethals,and related codes[J].IEEE Transactions on Information Theory,1994,40(2):301-319.
[2] J F Qian,L N Zhang,S X Zhu.(1+u) Constacyclic and cyclic codes over F₂+uF₂[J].Applied Mathematics Letters,2006,19(8):820-823.
[3] M C V Amarra,F R Nemenzo.On (1-u) cyclic codes over F_p^k+uF_p^k[J].Applied Mathematics Letters 21,2008,11:1129-1133.
[4] T Abular,I Siap.Constacyclic codes over F₂+uF₂[J].Journal of the Franklin Institute,2009,345:520-529.
[5] R Sobhani,E Esmaeili.Some constacyclic and cyclic codes over F_q[u]/< u^t+1>[J].IEICE Transactions on Foundamentals of Electronics,Communications and Computer Sciences,2010,93(4):808-813.
[6] X S Kai,S X Zhu,P Li.(1+λu) Constacyclic codes over F_p[u]/< u^k>[J].Journal of the Franklin Institute,2010,347:751-762.
[7] J F Qian.Cyclic codes over finite rings[A].Wireless Communications,Networking and Mobile Computing (WiCOM),7th International Conference[C].Wahan:IEEE Transations on Information Theory,2011.1-4.
[8] 王立启,朱士信.环F₂[u]/< u⁴>上的一类常循环码及其Gray象[J].电子与信息学报,2013,35(2):499-503. L Q Wang,S X Zhu.A class of constacyclic codes over F₂[u]/< u⁴> and its Gray image[J].Journal of Electronics and Information Technology,2013,35(2):499-503.(in Chinese)
[9] 施敏加,杨善林,朱士信.环F₂+uF₂上长为2^e的循环码的距离[J].电子学报,2011,39(1):29-34. Shi Min-jia,Yang Shan-lin,Zhu Shi-xin.On minimum distances of cyclic codes of length 2^e over F₂+uF₂[J].Acta Electronica Sinica,2011,39(1):29-34.(in Chinese)
[10] 施敏加.环F₂+uF₂+…+u^m-1F₂常循环自对偶码[J].电子学报,2013,41(6):1088-1092. Shi Min-jia.Constacyclic self-dual codes over ring F₂+uF₂+…+u^m-1F₂[J].Acta Electronica Sinica,2013,41(6):1088 -1092.(in Chinese)
[11] 施敏加,刘艳.环F_p+vF_p+v²F_p上线性码的各种重量的MacWilliams恒等式[J].电子学报,2014,42(7):1387-1391. Shi Min-jia,Liu Yan.Several weight enumerators of linear codes and their Macwilliams identities over ring F_p+vF_p+v²F_p[J].Acta Electronica Sinica,2014,42(7):1387-1391.(in Chinese)
[12] Minjia Shi.Optimal p-ary codes from constacyclic codes over F_p+vF_p[J].Chinese Journal of Electronics,2014,23(4):773-777.
[13] J Ding,H J Li .The Gray images of (1+u) constacyclic codes over F_2^m[u]/< u^k>[J].Journal of Applied Mathematics and Computing,2014,DOI 10.1007/s12190-014-0847-5.
[14] J Ding,H J Li .The Gray image of a class of constacyclic codes over polynomial residue rings[J].Journal of the Franklin Institute,2014,351:5467-5479.
[15] 李岩,朱士信.环F_p^m+uF_p^m+…+u^k－1F_p^m上的一类常循环码[J].合肥工业大学学报(自然科学版),2013,35(3):408-411. Y Li,S X Zhu.A class of constacyclic codes over the ring F_p^m+uF_p^m+…+u^k－1F_p^m[J].Journal of Hefei University of Technology,2013,35(3):408-411.(in Chinese)

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有限域上最优码的一种新的构造方法

A New Method on the Construction of Optimal Codes over Finite Field

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