AI 中文总结
研究在编码理论中，为一般秩度量码提出新的局部性定义，通过对码删截和缩短的研究，给出示例、构造及相关界，证明类似经典Tamo - Barg码的构造在此定义下最优。
AI 中文摘要
在编码理论中，局部恢复能够通过仅访问少量其他数据条目来高效恢复部分（丢失的）编码数据。局部性大多在汉明度量的背景下，针对单个符号的恢复进行了深入研究。在这项工作中，我们为一般的秩度量码提出了一种新的局部性定义。该定义与Kadhe、El Rouayheb、Duursma和Sprintson [IEEE 信息论汇刊，2019年] 的先前工作不同，它允许高效恢复支撑集中的任何元素，且不依赖于底层向量空间基的任何选择。我们的工作首先依赖于将码视为线性映射空间时对码删截和缩短的精确研究。然后我们给出示例和一般构造，展示我们的概念与Kadhe等人的概念之间的差异。接着我们推导出秩局部可恢复码的类似Singleton界，最后证明类似于经典Tamo - Barg码的一种构造相对于此界是最优的。
英文摘要
In coding theory, local recovery enables the efficient recovery of some part of (lost) coded data by accessing only a small number of other data entries. Locality was mostly but intensively studied for the recovery of individual symbols, that is, in the context of the Hamming metric. In this work, we propose a new definition of locality for general rank-metric codes. This definition differs from a previous work of Kadhe, El Rouayheb, Duursma and Sprintson [IEEE Trans. Inf. Theory 2019], by allowing to efficiently recover any element of the support, and without relying on any choice of bases of the underlying vector spaces. Our work firstly relies on a precise study of code puncturing and shortening for codes viewed as spaces of linear maps. We then provide examples and general constructions, showing the difference between our notion and that of Kadhe et al. We then derive a Singleton-like bound for rank locally recoverable codes, and we finally prove that a construction similar to classical Tamo-Barg codes is optimal with respect to this bound.
