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.

翻译：在这项工作中引入了仓储代码。 这些代码基本上被定义为最长的代码( 最高信息率的特高) 。 当字段扩展度为 1 美元时, 显示对齐的仓储代码的平位偏差值至少为 3 美元( 超一度校正), 对于固定的冗余 $ 美元, 基地规模 $ q 美元 和 外地扩展度度 $ 美元 。 ( 即 矩阵行数 ), 这些代码基本上被定义为: 最长的代码( 最短的 ), 最长的代码( 最高的信息率 ) 。 当字段扩展度为 1 美元 时, 显示 平坦的平整级平坦美代码 与最大大小 的平坦美分解码 。 在此情况下, 平坦的平坦坦调代码( 包括简单平坦的平坦的平坦版码 ), 以平坦的平坦的平坦式平坦的平坦代码, 以平坦的平坦的平坦的平坦的平坦的平坦的平坦代码 。 以平坦的平坦的平坦的平坦的平坦的平坦的平坦的平坦的平坦的平坦的平坦的平坦的平坦的平坦的平坦的平坦的平坦的平坦的平坦的平坦