Extension of Optimal Locally Repairable codes

Recent studies have delved into the construction of locally repairable codes (LRCs) with optimal minimum distance from function fields. In this paper, we present several novel constructions by extending the findings of optimally designed locally repairable codes documented in the literature. Let $C$ denote an optimal LRC of locality $r$, implying that every repairable block of $C$ is a $[r+1, r]$ MDS code, and $C$ maximizes its minimum distance. By extending a single coordinate of one of these blocks, we demonstrate that the resulting code remains an optimally designed locally repairable code. This suggests that the maximal length of an optimal LRC from rational function fields can be extended up to $q+2$ over a finite field $\mathbb{F}_q$. In addition, we give a new construction of optimal $(r, 3)$-LRC by extending one coordinate in each block within $C$. Furthermore, we propose a novel family of LRCs with Roth-Lempel type that are optimal under certain conditions. Finally, we explore optimal LRCs derived from elliptic function fields and extend a single coordinate of such codes. This approach leads us to confirm that the new codes are also optimal, thereby allowing their lengths to reach $q + 2\sqrt{q} - 2r - 2$ with locality $r$. We also consider the construction of optimal $(r, 3)$-LRC in elliptic function fields, with exploring one more condition.