Bounds on the minimum distance of locally recoverable codes

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.