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.

翻译：符号对码最初是为高密度存储系统中的应用而提出的，在该系统中无法读取单个符号。Yaakobi、Bruck和Siegel证明了二元线性循环码的最小配对距离满足$d_2 \geq \lceil 3d_H/2 \rceil$，并于2016年引入了$b$-符号度量码。本文研究了$b$-符号度量下的覆盖码。通过实例表明，Hamming度量下覆盖码的Delsarte界和Norse界在配对度量下并不成立。我们给出了线性码在$b$-符号度量下覆盖半径的冗余度界，并构造了达到该界的最优码。进一步证明了不存在最小配对距离为$7$的完美线性符号对码，且当$b\geq \frac{n+1}{2}$时不存在完美$b$-符号度量码。同时证明了大量循环码和代数几何码在$b$-符号度量下是非完美的。确定了Reed-Solomon码在$b$-符号度量下的覆盖半径。作为应用，还给出了列表可译$b$-符号度量码大小的广义Singleton界，并证明了广义MDS符号对码长度的上界。