2021/11/11 by Gerardo Vega, Vega, Gerardo, Félix Hernández +1
Computer Science · Social Sciences · #Coding theory and cryptography #Islamic Finance and Communication #Cooperative Communication and Network Coding
paper · pdf · doi:10.48550/arxiv.2111.06470
A class of optimal three-weight cyclic codes of dimension 3 over any finite\nfield was presented by Vega [Finite Fields Appl., 42 (2016) 23-38]. Shortly\nthereafter, Heng and Yue [IEEE Trans. Inf. Theory, 62(8) (2016) 4501-4513]\ngeneralized this result by presenting several classes of cyclic codes with\neither optimal three weights or a few weights. On the other hand, a class of\noptimal five-weight cyclic codes of dimension 4 over a prime field was recently\npresented by Li, et al. [Adv. Math. Commun., 13(1) (2019) 137-156]. One of the\npurposes of this work is to present a more general description for these\noptimal five-weight cyclic codes, which gives place to an enlarged class of\noptimal five-weight cyclic codes of dimension 4 over any finite field. As an\napplication of this enlarged class, we present the complete weight enumerator\nof a subclass of the optimal three-weight cyclic codes over any finite field\nthat were studied by Vega [Finite Fields Appl., 42 (2016) 23-38]. In addition,\nwe study the dual codes in this enlarged class of optimal five-weight cyclic\ncodes, and show that they are cyclic codes of length q2-1, dimension\nq2-5, and minimum Hamming distance 4. In fact, through several examples, we\nsee that those parameters are the best known parameters for linear codes.\n