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

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