Hamming and simplex codes for the sum-rank metric

• Published in: Designs, codes, and cryptography Vol. 88; no. 8; pp. 1521 - 1539
• Main Author: Martínez-Peñas, Umberto
• Format: Journal Article
• Language: English
• Published: New York Springer US 01.08.2020; Springer Nature B.V
• Subjects: Algorithms; Codes; Coding and Information Theory; Computer Science; Cryptology; Cryptology and Curves (in honour of Ruud Pellikaan); Decoding; Discrete Mathematics in Computer Science; Error correction; Hamming codes; Parameter estimation; Redundancy; Special Issue: Codes; Upper bounds; Hamming metric; Simplex codes; Sum-rank metric; 94B35; Rank metric; 94B05; Hamming codes; Locally repairable codes; 94B65; Multishot network coding
• ISSN: 0925-1022; 1573-7586
• DOI: 10.1007/s10623-020-00772-5

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.