几类射影三重和四重二元线性码及其应用
Several Classes of Projective Three-Weight and Four-Weight Binary Linear Codes and Their Applications
摘要: 本文通过恰当地选择定义集，构造了四类具有较少重量的射影二元线性码，并通过计算指数和确 定了它们的重量分布。 特别地，根据Grassl 的在线数据库，构造的某些二元线性码是最优的，并 且其中某些码的对偶是最优的或几乎最优的。 本文还利用Griesmer界刻画了四类线性码的最优 性，并得到了一些(几乎)距离最优的线性码或(近) Griesmer码。 在实际应用方面，本文得到的一 些线性码可以用于构造具有良好访问结构的秘密共享方案。
Abstract: In this paper, we construct four classes of projective binary linear codes with a few weights by selecting defining sets properly and completely determine their weight dis- tributions by calculating the exponential sums. Especially, some of the constructed binary linear codes are optimal according to the online Database of Grassl, and the duals of some of them are optimal or almost optimal. We also characterize the op- timality of these four classes of linear codes using the Griesmer bound and obtain some (almost) distance-optimal linear codes or (near) Griesmer codes. As applica- tions, some of the linear codes obtained in this paper can be used to construct secret sharing schemes with interesting access structures.
文章引用：周欢. 几类射影三重和四重二元线性码及其应用[J]. 应用数学进展, 2024, 13(8): 4064-4076. https://doi.org/10.12677/AAM.2024.138388

参考文献

[1] Qi, Y.F., Tang, C.M. and Huang, D.M. (2016) Binary Linear Codes with Few Weights. Finite Fields and Their Applications, 20, 208-211.
https://doi.org/10.1109/LCOMM.2015.2506576
[2] Yuan, J. and Ding, C. (2006) Secret Sharing Schemes from Three Classes of Linear Codes. IEEE Transactions on Information Theory, 52, 206-212.
https://doi.org/10.1109/TIT.2005.860412
[3] Calderbank, A.R. and Kantor, W.M. (1986) The Geometry of Two-Weight Codes. Bulletin of the London Mathematical Society, 18, 97-122.
https://doi.org/10.1112/blms/18.2.97
[4] Calderbank, A.R. and Goethals, J.M. (1984) Three-Weight Codes and Association Schemes. Philips Journal of Research, 39, 143-152.
[5] Ding, C., Helleseth, T., Klove, T. and Wang, X. (2007) A Generic Construction of Cartesian Authentication Codes. IEEE Transactions on Information Theory, 53, 2229-2235.
https://doi.org/10.1109/TIT.2007.896872
[6] Lidl, R. and Niederreiter, H. (1997) Finite Fields. Cambridge University Press.
https://doi.org/10.1017/CBO9780511525926
[7] Ding, K. and Ding, C. (2014) Binary Linear Codes with Three Weights. IEEE Communications Letters, 18, 1879-1882.
https://doi.org/10.1109/LCOMM.2014.2361516
[8] Ding, C. (2015) Linear Codes from Some 2-Designs. IEEE Transactions on Information Theory, 61, 3265-3275.
https://doi.org/10.1109/TIT.2015.2420118
[9] Ding, K. and Ding, C. (2015) A Class of Two-Weight and Three-Weight Codes and Their Applications in Secret Sharing. IEEE Transactions on Information Theory, 61, 5835-5842.
https://doi.org/10.1109/TIT.2015.2473861
[10] Ding, C., Li, C., Li, N. and Zhou, Z. (2016) Three-Weight Cyclic Codes and Their Weight Distributions. Discrete Mathematics, 339, 415-427.
https://doi.org/10.1016/j.disc.2015.09.001
[11] Jian, G., Feng, R. and Wu, H. (2017) Generalized Hamming Weights of Three Classes of Linear Codes. Finite Fields and Their Applications, 45, 341-354.
https://doi.org/10.1016/j.ffa.2017.01.001
[12] Li, F. (2018) A Class of Cyclotomic Linear Codes and Their Generalized Hamming Weights. Applicable Algebra in Engineering, Communication and Computing, 29, 501-511.
https://doi.org/10.1007/s00200-018-0355-1
[13] Li, F. and Li, X. (2021) Weight Distributions and Weight Hierarchies of Two Classes of Binary Linear Codes. Finite Fields and Their Applications, 73, Article 101865.
https://doi.org/10.1016/j.ffa.2021.101865
[14] Liu, Z.H. and Wang, J.L. (2020) Notes on Generalized Hamming Weights of Some Classes of Binary Codes. Cryptography and Communications, 12, 645-657.
https://doi.org/10.1007/s12095-019-00404-3
[15] Moisio, M. (2009) Explicit Evaluation of Some Exponential Sums. Finite Fields and Their Applications, 15, 644-651.
https://doi.org/10.1016/j.ffa.2009.05.005
[16] Wang, Q.Y., Ding, K.L. and Xue, R. (2015) Binary Linear Codes with Two Weights. IEEE Communications Letters, 19, 1097-1100.
https://doi.org/10.1109/LCOMM.2015.2431253
[17] Ashikhmin, A. and Barg, A. (1998) Minimal Vectors in Linear Codes. IEEE Transactions on Information Theory, 44, 2010-2017.
https://doi.org/10.1109/18.705584
[18] Huffman, W.C. and Pless, V. (2003) Fundamentals of Error-Correcting Codes. Cambridge University Press.
https://doi.org/10.1017/CBO9780511807077
[19] Ding, C. (2015) Codes from Difference Sets. World Scientific.
https://doi.org/10.1142/9283
[20] Ding, C. (2018) Designs from Linear Codes. World Scientific.
https://doi.org/10.1142/11101
[21] Heng, Z., Ding, C. and Wang, W. (2020) Optimal Binary Linear Codes from Maximal Arcs. IEEE Transactions on Information Theory, 66, 5387-5394.
https://doi.org/10.1109/TIT.2020.2970405
[22] Heng, Z., Wang, Q. and Ding, C. (2020) Two Families of Optimal Linear Codes and Their Subfield Codes. IEEE Transactions on Information Theory, 66, 6872-6883.
https://doi.org/10.1109/TIT.2020.3006846
[23] Hyun, J.Y., Kim, H.K., Wu, Y. and Yue, Q. (2020) Optimal Minimal Linear Codes from Posets. Designs, Codes and Cryptography, 88, 2475-2492.
https://doi.org/10.1007/s10623-020-00793-0
[24] Hyun, J.Y., Lee, J. and Lee, Y. (2020) Infinite Families of Optimal Linear Codes Constructed from Simplicial Complexes. IEEE Transactions on Information Theory, 66, 6762-6773.
https://doi.org/10.1109/TIT.2020.2993179
[25] Wang, X., Zheng, D. and Ding, C. (2021) Some Punctured Codes of Several Families of Binary Linear Codes. IEEE Transactions on Information Theory, 67, 5133-5148.
https://doi.org/10.1109/TIT.2021.3088146
[26] Hu, Z., Li, N., Zeng, X., Wang, L. and Tang, X. (2022) A Subfield-Based Construction of Optimal Linear Codes over Finite Fields. IEEE Transactions on Information Theory, 68, 4408-4421.
https://doi.org/10.1109/TIT.2022.3163651
[27] Zhu, C.Z. and Liao, Q.Y. (2022) Several Classes of Projective Few-Weight Linear Codes and Their Applications. arXiv:2211.04519
[28] Grassl M. (2024) Bounds on the Minimum Distance of Linear Codes and Quantum Codes. http://www.codetables.de

为你推荐