Access-optimal MSR Codes with Optimal Sub-packetization over Small Fields

This paper presents a new construction of access-optimal minimum storage regenerating codes which attain the sub-packetization bound. These distributed storage codes provide the minimum storage in a node, and in addition have the following two important properties: first, a helper node accesses the minimum number of its symbols for repair of a failed node; second, 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). The goal of this paper is to provide a construction of such optimal codes over the smallest possible finite fields. Our construction is based on perfect matchings of complete graphs and hypergraphs, and on a rational canonical form of matrices which constitute a generator matrix of the constructed codes. The field size required for our construction is significantly smaller when compared to previously known codes.

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