Alphabet-dependent upper bounds for locally repairable codes with joint locality

In this paper, we propose two upper bounds for locally repairable codes (LRCs) with joint locality. The bounds take into account the code length, dimension, minimum distance, joint locality, and field size, i.e., the size of code alphabet. We also introduce binary linear LRCs designed by utilizing simple graphs. Some of them with certain choice of parameters achieve both of the bounds simultaneously with equality. With the codes, we show that our bounds are tighter than existing alphabet-dependent upper bound.