Optimal binary and ternary locally repairable codes with minimum distance 6

A locally repairable code (LRC) is a code that can recover any symbol of a codeword by reading at most r other symbols, denoted by r-LRC. In this paper, we study binary and ternary linear LRCs with disjoint repair groups and minimum distance d = 6. Using the intersection subspaces technique, we explicitly construct dimensional optimal LRCs. First, based on the intersection subspaces constructed by t-spread, a construction of binary LRCs is designed. Particularly, a class of binary linear LRCs with r = 11 is optimal in terms of achieving a sphere-packing type upper bound. Next, by using the Kronecker product of two matrices, two classes of dimensional optimal ternary LRCs with small locality (r = 3, 5) are presented. Compared to previous results, our construction is more flexible regarding code parameters. Finally, we also discuss the parameters of a code obtained by applying a shortening operation to our LRCs. We show that these shortened LRCs are also k-optimal.