Griesmer-Optimal Locally Repairable Codes via Lengthening Reed-Muller Codes

Locally repairable codes (LRCs) have attracted considerable attention in recent years due to their practical applications in distributed storage systems and their inherent theoretical significance. In this paper, we focus on alphabet-dependent optimal LRCs. We first establish some new bounds for the code length, dimension and minimum distance of $(r, \delta)$-LRCs, respectively. In particular, we propose an explicit bound, the Griesmer-like bound, which extends the classical Griesmer bound to accommodate locality constraints. This is also the generalization of a Griesmer-like bound proposed by Hao et al. in 2020. Furthermore, by lengthening the $q$-ary first-order Reed-Muller code, we present two classes of (2, $q$) -LRCs that achieve this newly derived Griesmer-like bound. It is worth noting that the first class of $(2, q)$-LRCs does not meet the classical Griesmer bound as linear codes. To the best of our knowledge, it is the first class of Griesmer-like LRCs that are not Griesmer codes. This also implies that our Griesmer-like bound for $(r, \delta)$-LRCs is non-trivial.