Optimal quaternary $$(r,\delta )$$-locally recoverable codes: their structures and complete classification

Abstract

Locally recoverable codes (LRCs) have been introduced as a family of erasure codes that support the repair of a failed storage node by contacting a small number of other nodes in the cluster. Boosted by their applications in distributed storage, LRCs have attracted a lot of attention in recent literature since the concept of codes with locality r was introduced by Gopalan et al. in 2012. Aiming to recover the data from several concurrent node failures, the concept of r-locality was later generalized as $$(r, \delta )$$ -locality by Prakash et al. An $$(r, \delta )$$ -LRCs in which every code symbol has $$(r, \delta )$$ -locality is said to be optimal if it achieves the Singleton-like bound with equality. In present paper, we are interested in optimal $$(r, \delta )$$ -LRCs over small fields, more precisely, over quaternary field. We study their parity-check matrices or generator matrices, using the properties of projective space. The classification of optimal quaternary $$(r,\delta )$$ -LRCs and their explicit code constructions are proposed by examining all possible parameters.