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

An <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$(r,\ \delta)$</tex> -locally repairable code (LRC) is an <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$[n,\ k,\ d]$</tex> linear code that permits the reconstruction of each code symbol by accessing up to <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$r$</tex> other symbols in the event of at most <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$\delta-1$</tex> erasures. In this paper, by characterizing the weight type hierarchy of codewords, we offer explicit expressions of the weight distributions for q-ary optimal <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$(r=2,\ \delta)$</tex> -LRCs with minimum distance <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$2\delta+1$</tex> and even code dimension, as well as for <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$(r,\ \delta)$</tex> -LRCs with minimum distance <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$\delta+1$</tex> under the condition that <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$k\geq 5r-1$</tex>. These corresponding parameter conditions ensure all the <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$(r,\ \delta)$</tex> -LRCs we studied possess disjoint locality groups. Furthermore, we demonstrate that the weight distributions of optimal <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$(r=2,\ \delta)$</tex> -LRCs can be uniquely determined only for specific minimum distances of <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$\delta, \delta+1$</tex> or <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$2\delta+1$</tex>.