Constructions of High-Rate MSR Codes over Small Fields

A novel technique for construction of minimum storage regenerating (MSR) codes is presented. Based on this technique, three explicit constructions of MSR codes are given. The first two constructions provide access-optimal MSR codes, with two and three parities, respectively, which attain the sub-packetization bound for access-optimal codes. The third construction provides longer MSR codes with three parities, which are not access-optimal, and do not necessarily attain the sub-packetization bound. In addition to a minimum storage in a node, all three constructions allow the entire data to be recovered from a minimal number of storage nodes. That is, given storage $\ell$ in each node, the entire stored data can be recovered from any $2\log2 \ell$ for 2 parity nodes, and either $3\log3\ell$ or $4\log3\ell$ for 3 parity nodes. Second, in the first two constructions, a helper node accesses the minimum number of its symbols for repair of a failed node (access-optimality). The generator matrix of these codes is based on perfect matchings of complete graphs and hypergraphs, and on a rational canonical form of matrices. The goal of this paper is to provide a construction of such optimal codes over the smallest possible finite fields. For two parities, the field size is reduced by a factor of two for access-optimal codes compared to previous constructions. For three parities, in the first construction the field size is $6\log3 \ell+1$ (or $3\log3 \ell+1$ for fields with characteristic 2), and in the second construction the field size is larger, yet linear in $\log3\ell$. Both constructions with 3 parities provide a significant improvement over existing previous works, since only non-explicit constructions with exponential field size (in $\log_3\ell$) were known so far.