Covering b-Symbol Metric Codes and the Generalized Singleton Bound [article]

Hao Chen
2022 arXiv   pre-print
Symbol-pair codes were proposed for the application in high density storage systems, where it is not possible to read individual symbols. Yaakobi, Bruck and Siegel proved that the minimum pair-distance of binary linear cyclic codes satisfies ⌈ d_2 ≥3d_H/2⌉ and introduced b-symbol metric codes in 2016. In this paper covering codes in b-symbol metrics are considered. Some examples are given to show that the Delsarte bound and the Norse bound for covering codes in the Hamming metric are not true
more » ... r covering codes in the pair metric. We give the redundancy bound on covering radii of linear codes in the b-symbol metric and give some optimal codes attaining this bound. Then we prove that there is no perfect linear symbol-pair code with the minimum pair distance 7 and there is no perfect b-symbol metric code if b≥n+1/2. Moreover a lot of cyclic and algebraic-geometric codes are proved non-perfect in the b-symbol metric. The covering radius of the Reed-Solomon code as a b-symbol code is determined. As an application the generalized Singleton bound on the sizes of list-decodable b-symbol codes is also presented. Then an upper bound on lengths of general MDS symbol-pair codes is proved.
arXiv:2206.12668v1 fatcat:ptda5ef3wzfbngc72y2m66xi2m