Optimal binary locally repairable codes with joint information locality

We propose new families of optimal binary locally repairable codes, called crown codes and ring codes, that achieve optimal code rate for joint information locality (r₁, r₂)i = (2, 3) and (r₁, r₂)i = (2, 4), respectively. The joint information locality is a set of numbers of symbols for repairing various erasure patterns of information symbols. We simplify the code design problem utilizing simple graphs, and present that special structures of the graphs provide good locality properties.