Bounds on the minimum distance of locally recoverable codes

2023/12/31 by Kurz, Sascha
#94B05 #94B27 #Combinatorics (math.CO) #Discrete Mathematics (cs.DM) #FOS: Computer and information sciences #FOS: Mathematics

paper · doi:10.48550/arxiv.2401.00418

Abstract

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 n₂(k,d,2) and for k≤ 6 we exactly determine all values of n₂(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.

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