Some New Sequential-Recovery LRCs Based on Good Polynomials

We propose a new construction of sequential-recovery Locally Repairable Codes (LRCs) of length
$n$
with even locality
$r$
for two erasures, based on some ‘good’ polynomials, over a relatively small alphabet of size
$q \approx \frac {(r+1)n}{r+2}$
, which becomes rate optimal in some cases. We also derive an explicit form of the upper bound on the minimum distance of these codes with some additional constraints. The minimum distance of the proposed sequential-recovery LRCs for
$r=2$
achieves this explicit bound when
$k = \frac {n}{2}$
and is one less than the bound when
$k < \frac {n}{2}$
.