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Symbol-pair codes were proposed for the application in high density storagesystems, where it is not possible to read individual symbols. Yaakobi, Bruckand Siegel proved that the minimum pair-distance of binary linear cyclic codessatisfies $d_2 \geq \lceil 3d_H/2 \rceil$ and introduced $b$-symbol metriccodes in 2016. In this paper covering codes in $b$-symbol metrics areconsidered. Some examples are given to show that the Delsarte bound and theNorse bound for covering codes in the Hamming metric are not true for coveringcodes in the pair metric. We give the redundancy bound on covering radius oflinear codes in the $b$-symbol metric and give some optimal codes attainingthis bound. Then we prove that there is no perfect linear symbol-pair code withthe 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 codesare proved non-perfect in the $b$-symbol metric. The covering radius of theReed-Solomon code in the $b$-symbol metric is determined. As an application thegeneralized Singleton bound on the sizes of list-decodable $b$-symbol metriccodes is also presented. Then an upper bound on lengths of general MDSsymbol-pair codes is proved.

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