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≤ 7 we exactly determine all values of n_2(k,d,2) and for k≤ 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.