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

n (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 kα , 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.