Hypergraph-Based Binary Locally Repairable Codes With Availability

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.