Constructions of High-Rate MSR Codes over Small Fields

Three constructions of minimum storage regenerating (MSR) codes are presented. 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 larger 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, these codes have the following two important properties: first, given storage l in each node, the entire stored data can be recovered from any 2 log l (any 3 log l) for 2 parity nodes (for 3 parity nodes, respectively); second, for the first two constructions, a helper node accesses the minimum number of its symbols for repair of a failed node (access-optimality). The goal of this paper is to provide a construction of such optimal codes over the smallest possible finite fields. 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 field size required for our construction is significantly smaller when compared to previously known codes.