Some Results on the Improved Bound and Construction of Optimal $(r,\delta)$ LRCs

TL;DR

This paper advances the understanding of optimal $(r,)$ locally repairable codes by improving bounds on maximum code length, providing geometric characterizations, and constructing new codes with better parameters.

Contribution

It improves upper bounds on the maximum length of optimal $(r,)$ LRCs and offers a geometric characterization and new constructions based on sunflower structures.

Findings

Improved upper bounds for code length when $2+1  d  2+2.

Complete geometric characterization of optimal $(r=2,)$-LRCs in $PG(2,q)$.

New constructions of optimal $(r,)$ LRCs outperform previous methods.

Abstract

Locally repairable codes (LRCs) with $(r,\delta)$ locality were introduced by Prakash \emph{et al.} into distributed storage systems (DSSs) due to their benefit of locally repairing at least $\delta-1$ erasures via other $r$ survival nodes among the same local group. An LRC achieving the $(r,\delta)$ Singleton-type bound is called an optimal $(r,\delta)$ LRC. Constructions of optimal $(r,\delta)$ LRCs with longer code length and determining the maximal code length have been an important research direction in coding theory in recent years. In this paper, we conduct further research on the improvement of maximum code length of optimal $(r,\delta)$ LRCs. For $2\delta+1\leq d\leq 2\delta+2$, our upper bounds largely improve the ones by Cai \emph{et al.}, which are tight in some special cases. Moreover, we generalize the results of Chen \emph{et al.} and obtain a complete characterization of optimal $(r=2, \delta)$-LRCs in the sense of geometrical existence in the finite projective plane $PG(2,q)$. Within this geometrical characterization, we construct a class of optimal $(r,\delta)$ LRCs based on the sunflower structure. Both the construction and upper bounds are better than previous ones.