We consider locally recoverable codes (LRCs) and aim to determine the smallest possible length $n=n_q(k,d,r)$ of a linear $[n,k,d]_q$-code with locality $r$. For $k\le 7$ we exactly determine all values of $n_2(k,d,2)$ and for $k\le 6$ we exactly determine all values of $n_2(k,d,1)$. For the ternary field we also state a few numerical results. As a general result we prove that $n_q(k,d,r)$ equals the Griesmer bound if the minimum Hamming distance $d$ is sufficiently large and all other parameters are fixed.

翻译：我们考虑局部可修复码（LRC），旨在确定具有局部性r的线性[n,k,d]_q-码的最小可能长度n=n_q(k,d,r)。当k≤7时，我们精确确定了n_2(k,d,2)的所有取值；当k≤6时，精确确定了n_2(k,d,1)的所有取值。在三元域上，我们也给出了一些数值结果。作为一般性结论，我们证明：当最小汉明距离d充分大且其余参数固定时，n_q(k,d,r)等于Griesmer界。