Binary locally repairable codes from complete multipartite graphs

In this paper, we propose a construction of binary locally repairable codes from complete multipartite graphs. Our codes possess joint locality (r₁ = 2, r₂ = 3 or 4). Joint locality is a set of numbers of symbols for repairing various erasure patterns of symbols. We also provide availability properties of our codes. A code is said to have (r, t)-availability if each of its symbols can be repaired by t disjoint groups of other symbols, each of a size of at most r. As a meaningful result, the proposed code construction can generate binary locally repairable codes achieving (2, t)-availability for any positive integer t.