Sum-rank Hamming codes are introduced in this work. They are essentially defined as the longest codes (thus of highest information rate) with minimum sum-rank distance at least 3 (thus one-error-correcting) for a fixed redundancy r , base-field size q and field-extension degree m (i.e., number of matrix rows). General upper bounds on their code length, number of shots or sublengths and average sublength are obtained based on such parameters. When the field-extension degree is 1, it is shown that sum-rank isometry classes of sum-rank Hamming codes are in bijective correspondence with maximal-size partial spreads. In that case, it is also shown that sum-rank Hamming codes are perfect codes for the sum-rank metric. Also in that case, estimates on the parameters (lengths and number of shots) of sum-rank Hamming codes are given, together with an efficient syndrome decoding algorithm. Duals of sum-rank Hamming codes, called sum-rank simplex codes, are then introduced. Bounds on the minimum sum-rank distance of sum-rank simplex codes are given based on known bounds on the size of partial spreads. As applications, sum-rank Hamming codes are proposed for error correction in multishot matrix-multiplicative channels and to construct locally repairable codes over small fields, including binary.

中文翻译：

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和秩度量的汉明码和单纯形码

在这项工作中引入了和秩汉明码。它们本质上被定义为最长的代码（因此具有最高的信息速率），对于固定的冗余 r 、基场大小 q 和场扩展度 m（即矩阵行数）。基于这些参数获得它们的代码长度、镜头数或子长度以及平均子长度的一般上限。当场扩展度为1时，表明和秩汉明码的和秩等距类与最大尺寸部分展开呈双射对应关系。在那种情况下，还表明和秩汉明码是和秩度量的完美代码。同样在这种情况下，给出了和秩汉明码的参数（长度和发射次数）的估计值，连同有效的综合症解码算法。然后引入了和秩汉明码的对偶，称为和秩单纯形码。和秩单纯形码的最小和秩距离的界限是基于部分扩展大小的已知界限给出的。作为应用，提出了和秩汉明码用于多镜头矩阵乘法通道中的纠错，并在包括二进制在内的小域上构建可局部修复的代码。