Hypergraph-Based Binary Locally Repairable Codes With Availability

Abstract

We study a hypergraph-based code construction for binary locally repairable codes (LRCs) with availability. A symbol of a code is said to have $(r, t)$ -availability if it can be recovered from $t$ disjoint repair sets of other symbols, each set of size at most $r$ . We refer a systematic code to an LRC with $(r, t)_{i}$ -availability if its information symbols have $(r, t)$ -availability and a code to an LRC with $(r, t)_{a}$ -availability if its all symbols have $(r, t)$ -availability. We construct binary LRCs with $(r, t)_{i}$ -availability from linear $r$ -uniform $t$ -regular hypergraphs. As a special case, we also construct binary LRCs with $(r, t)_{a}$ -availability from labeled linear $r$ -uniform $t$ -regular hypergraphs. Moreover, we extend the hypergraph-based codes to increase the minimum distance. All the proposed codes achieve a well-known Singleton-like bound with equality.