论文标题
覆盖$ b $ -Symbol公制代码和广义单例绑定
Covering $b$-Symbol Metric Codes and the Generalized Singleton Bound
论文作者
论文摘要
为高密度存储系统中的应用提供了符号对代码，在此不可能读取单个符号。 Yaakobi，Bruck和Siegel证明了二进制线性循环代码的最低配对距离满足$ d_2 \ geq \ geq \ lceil 3D_H/2 \ rceil $，并引入了$ b $ -Symbol Metrices codes，2016年。在$ b $ -symbol -bol-bor-y-ymbol-bolt actrics中，涵盖了本文中的代码。给出了一些示例，以表明Delsarte Bonding和用于覆盖锤量指标代码的北欧绑定对于覆盖对度量指标的代码并不正确。我们在$ b $ -Symbol公制中覆盖线性代码半径的冗余界限，并给出一些最佳代码，以达到此界限。然后，我们证明没有最小对距离$ 7 $的完美线性符号代码，如果$ b \ geq \ frac {n+1} {2} $，则没有完美的$ b $ -symbol公制代码。此外，在$ B $ -Symbol公制中，许多循环和代数几何代码被证明是不完美的。确定了$ b $ symbol公制中的芦苇 - 固体代码的覆盖半径。作为应用程序，还介绍了列表可调$ b $ -symbol公制代码的大小的广义单例。然后证明了一般MDS符号对代码长度的上限。
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.
