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.

本地可修复码（LRC）是通过读取最多\(r\)个其他符号来恢复码字的任何符号的代码，记为\(r\)-LRC。在本文中，我们研究了具有不相交修复组和最小距离 \(d \) = 6 的二元和三元线性 LRC。使用相交子空间技术，我们显式地构造了维度最优 LRC。首先，基于\(t \)-spread构造的交集子空间，设计了二值LRC的构造。特别是，一类具有 \(r \) = 11 的二元线性 LRC 在实现球堆积型上限方面是最佳的。接下来，通过使用两个矩阵的 Kronecker 积，提出了两类具有小局部性 (\(r \) = 3, 5) 的维数最优三元 LRC。与之前的结果相比，我们的构造在代码参数方面更加灵活。最后，我们还讨论了通过对 LRC 应用缩短操作而获得的代码的参数。我们证明这些缩短的 LRC 也是 k 最优的。