有限环F_2+F_2u+F_2u~2+F_2u~3上的线性码
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摘要
研究了有限环F_2+F_2u+F_2u~2+F_2u~3上的线性码结构。定义了环F_2+F_2u+F_2u~2+F_2u~3上的线性码的Lee重量、Hamming重量、完全重量及对称重量计数器的概念,利用有限域上线性码和Gray映射的性质,给出了本环上的线性码之间的各重量分布的Macwilliams恒等式。最后,得到一类最优的线性码。
This paper studies the structure of the linear code over finite ring F_2+counting formulas of Lee weight, Hamming weight, and complete weight and symmetrized weight of linear codes over ring F_2+F_2 u +F_2 u~2+F_2 u~3. By the relationship of linear codes and the proposition of the Gray map, the paper give the Macwilliams identities between the linear codes. The study also obtains a class optimal linear codes.
This paper studies the structure of the linear code over finite ring F_2+counting formulas of Lee weight, Hamming weight, and complete weight and symmetrized weight of linear codes over ring F_2+F_2 u +F_2 u~2+F_2 u~3. By the relationship of linear codes and the proposition of the Gray map, the paper give the Macwilliams identities between the linear codes. The study also obtains a class optimal linear codes.
引文
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[3]K. Shiromoto. A basic exact sequence for the Lee and Eu?clidean weights of linear codes overZ4[J].Linear Algebraand its Applications,1999,295(1-3):191-2000.
[4]Z.X.Wan. The Macwilliams identity of for linear codes overGalois rings[A]. Numbers Information and Complexity[C].Netherlands:Kluwer Academic Publishers,2000:333-338.
[5]施敏加,杨善林,朱士信.F2+uF2上长为2s的循环码的距离[J].电子学报,2011,39(1):29-34.
[6]余海峰,朱士信.环F2+uF2上线性码及其对偶码的Mac?williams恒等式[J].中国科学技术大学学报,2006,36(2):29-34.
[7] M. Shi, D. Wang, P. Sole′. Linear codesoverF3+uF3+u2F3:MacWilliams identities,optimal terna?ry codes from one-Lee weight codes and two-Lee weightcodes[J]. Journal of Applied Mathematics and Computing,2016,51(1-2):1-18.
[2]A.Ashikhmin. On generalized Hamming weights for Galoisring linear codes[J]. Designs,Codes and Cryptography,1998,14(2):107-126.
[3]K. Shiromoto. A basic exact sequence for the Lee and Eu?clidean weights of linear codes overZ4[J].Linear Algebraand its Applications,1999,295(1-3):191-2000.
[4]Z.X.Wan. The Macwilliams identity of for linear codes overGalois rings[A]. Numbers Information and Complexity[C].Netherlands:Kluwer Academic Publishers,2000:333-338.
[5]施敏加,杨善林,朱士信.F2+uF2上长为2s的循环码的距离[J].电子学报,2011,39(1):29-34.
[6]余海峰,朱士信.环F2+uF2上线性码及其对偶码的Mac?williams恒等式[J].中国科学技术大学学报,2006,36(2):29-34.
[7] M. Shi, D. Wang, P. Sole′. Linear codesoverF3+uF3+u2F3:MacWilliams identities,optimal terna?ry codes from one-Lee weight codes and two-Lee weightcodes[J]. Journal of Applied Mathematics and Computing,2016,51(1-2):1-18.