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.