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 $\lceil d_2 \geq \frac{3d_H}{2}\rceil$ 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 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 $b\geq \frac{n+1}{2}$. 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 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.

翻译：在高密度储存系统中,提出了用于高密度储存系统中应用的符号代码。 Yaakobi、Breck和Siegel证明,双线线性环曲码的最低配对距离符合$\lceil d_2\geq\frac{3d_H ⁇ 2 ⁇ rceil$,并在2016年引入了$b$-symbol 公标代码。本文中考虑的是以美元表示的代号为$b$-symbol 的代号。提供了一些实例,以表明在Hamming 公标中,受约束的代号和受约束的代号并非包含的代号。我们给双线性线性线性码的最小配对时间($d_2\ geq) d_2\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\