最佳三加权循环编码子类的完整重量量计器 (The complete weight enumerator of a subclass of optimal three-weight cyclic codes)

A class of optimal three-weight cyclic codes of dimension 3 over any finite field was presented by Vega [Finite Fields Appl., 42 (2016) 23-38]. Shortly thereafter, Heng and Yue [IEEE Trans. Inf. Theory, 62(8) (2016) 4501-4513] generalized this result by presenting several classes of cyclic codes with either optimal three weights or a few weights. On the other hand, a class of optimal five-weight cyclic codes of dimension 4 over a prime field was recently presented by Li, et al. [Adv. Math. Commun., 13(1) (2019) 137-156]. One of the purposes of this work is to present a more general description for these optimal five-weight cyclic codes, which gives place to an enlarged class of optimal five-weight cyclic codes of dimension 4 over any finite field. As an application of this enlarged class, we present the complete weight enumerator of a subclass of the optimal three-weight cyclic codes over any finite field that were studied by Vega [Finite Fields Appl., 42 (2016) 23-38]. In addition, we study the dual codes in this enlarged class of optimal five-weight cyclic codes, and show that they are cyclic codes of length $q^2-1$, dimension $q^2-5$, and minimum Hamming distance 4. In fact, through several examples, we see that those parameters are the best known parameters for linear codes.

翻译：Vega[Finite Fields Appl., 42/2016 23-38] 介绍了关于任何有限字段的3个方面的最佳三重周期代码。随后不久,Heng和Yue[IEEE Trans. Inf. Theory, 62(8) & 4501-4513] 介绍了关于任何有限字段的3级最佳三重周期代码。另一方面,Li等人等人(Adv. Math.Commun., 13(1) (2019) 137-156) 介绍了关于任何有限字段的4级最佳三重周期代码。这项工作的目的之一是更全面地描述这些最佳5重周期代码。 62(8) 4501-4513, 51313, 513,1,1,1,1,1,1,1,1,1,1,1,1,2,1,2,2,2,3,2,2,3,2,3,2,2,2,2,3,2,2,2,3,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2