On the Construction of Local Parities for $(r, t)$ -Availability in Distributed Storage

Motivated by applications involving hot data, the notion of $ {(r, t)}$ -availability was introduced, where a symbol is said to have $ {(r, t)}$ -availability if it can be reconstructed, respectively, from $ {t}$ disjoint repair alternatives of other symbols, each of size at most $ {r}$ . The key to constructing a family of locally repairable codes with $ {(r, t)}$ -availability is the design of a membership matrix for dividing global parities into local ones such that each repair group contains only one parity symbol. Although explicit designs of membership matrices are available, it remains unclear whether membership matrices 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 the design of membership matrices and the design of a combinatorial object in the well-known combinatorial design theory, called resolvable configurations. This paper then proposes several direct designs of membership matrices based on Euclidean geometry, finite fields, circulant permutation matrices, affine permutation matrices, and Reed–Solomon codes. By leveraging the notion of resolvable configurations and the Kronecker product, this paper further proposes a two-step design of membership matrices. The proposed direct and two-step designs of membership matrices are shown to have significantly better parameters than those recently available.