Multilinear algebra for minimum storage regenerating codes: a generalization of the product-matrix construction

• Published in: Applicable algebra in engineering, communication and computing Vol. 34; no. 4; pp. 717 - 743
• Main Authors: Duursma, Iwan, Wang, Hsin-Po
• Format: Journal Article
• Language: English
• Published: Berlin/Heidelberg Springer Berlin Heidelberg 01.07.2023; Springer Nature B.V
• Subjects: Artificial Intelligence; Computer Hardware; Computer Science; Lower bounds; Mathematical analysis; Matrices (mathematics); Nodes; Original Paper; Symbolic and Algebraic Manipulation; Symbols; Theory of Computation; Regenerating code; Multilinear algebra; 94B27; 15A69; Minimum storage regeneration; Distributed storage system; 68P20
• ISSN: 0938-1279; 1432-0622
• DOI: 10.1007/s00200-021-00526-3

An ( n , k , d , α ) -MSR (minimum storage regeneration) code is a set of n nodes used to store a file. For a file of total size k α , each node stores α symbols, any k nodes determine the file, and any d nodes can repair any other node by each sending out α / ( d - k + 1 ) symbols. In this work, we express the product-matrix construction of ( n , k , 2 ( k - 1 ) , k - 1 ) -MSR codes in terms of symmetric algebras. We then generalize the product-matrix construction to ( n , k , ( k - 1 ) t t - 1 , k - 1 t - 1 ) -MSR codes for general t ⩾ 2 , while the t = 2 case recovers the product-matrix construction. Our codes’ sub-packetization level— α —is small and independent of n . It is less than L 2.8 ( d - k + 1 ) , where L is Alrabiah–Guruswami’s lower bound on  α . Furthermore, it is less than other MSR codes’ α for a set of practical parameters. Finally, we discuss how our code repairs multiple failures at once.