Weight Distributions of Two Classes of Optimal $(r, \delta)$-Locally Repairable Codes

An $(r,\ \delta)$ -locally repairable code (LRC) is an $[n,\ k,\ d]$ linear code that permits the reconstruction of each code symbol by accessing up to $r$ other symbols in the event of at most $\delta-1$ erasures. In this paper, by characterizing the weight type hierarchy of codewords, we offer explicit expressions of the weight distributions for q-ary optimal $(r=2,\ \delta)$ -LRCs with minimum distance $2\delta+1$ and even code dimension, as well as for $(r,\ \delta)$ -LRCs with minimum distance $\delta+1$ under the condition that $k\geq 5r-1$. These corresponding parameter conditions ensure all the $(r,\ \delta)$ -LRCs we studied possess disjoint locality groups. Furthermore, we demonstrate that the weight distributions of optimal $(r=2,\ \delta)$ -LRCs can be uniquely determined only for specific minimum distances of $\delta, \delta+1$ or $2\delta+1$.