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

Locally repairable codes (LRCs) with
$(r,\ \delta)$
locality were introduced by Prakash 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 whose minimum distance
$d$
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 letter, 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 et al., which are tight in some special cases. Moreover, we generalize the results of Chen 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 new class of optimal
$(r,\ \delta)$
LRCs based on the sunflower structure.