Design of membership matrices for (r, t)-availability in distributed storage

This paper is concerned with the construction of local parities for optimal locally repairable codes (LRCs) with (r, t)-availability in distributed storage, where a symbol is said to have (r, t)-availability if it can be reconstructed from t disjoint repair alternatives of other symbols, each of size at most r. The key to constructing a family of optimal LRCs with (r, t)-availability that can support a scaling number of parallel reads while keeping the rate to be an arbitrarily high constant is the design of a (0,1)-matrix R, called a membership matrix, for dividing global parities into local ones. Although explicit designs of R are available, it remains unclear whether R with significantly better parameters can be constructed, which was a question left open. To tackle the open problem, this paper first provides a connection between designs of R and a combinatorial object in the well-known combinatorial design theory, called resolvable configurations. This paper then proposes several designs of R based respectively on Euclidean geometry, circulant permutation matrices, and affine permutation matrices, which, to the best knowledge of the author of this paper, are also new to resolvable configurations. The proposed designs of R are shown to have significantly better parameters than those in the literature.