Covering $b$-Symbol Metric Codes and the Generalized Singleton Bound

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 $d_2 \geq \lceil 3d_H/2 \rceil$ 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 for covering codes in the pair metric. We give the redundancy bound on covering radius 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 $b\geq \frac{n+1}{2}$. 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 in the $b$-symbol metric is determined. As an application the generalized Singleton bound on the sizes of list-decodable $b$-symbol metric codes is also presented. Then an upper bound on lengths of general MDS symbol-pair codes is proved.