# New Constructions of Optimal $(r,\delta)$-LRCs via Algebraic Function Fields

## TL;DR

This paper introduces new algebraic geometric methods for constructing optimal locally repairable codes with flexible minimum distances, achieving some of the longest known code lengths for such parameters.

## Contribution

It extends existing frameworks using elliptic, hyperelliptic, and superelliptic curves to construct explicit optimal $(r,\,\delta)$-LRCs with lengths approaching theoretical bounds.

## Key findings

- Constructed explicit optimal $(r,3)$-LRCs with lengths near $q + 2\sqrt{q}$.
- Developed families of optimal $(r,3)$-LRCs and $(g+1-g',g+1+g')$-LRCs with lengths up to $q + 2g\sqrt{q}$.
- Achieved some of the longest known code lengths among existing constructions with flexible minimum distances.

## Abstract

Constructing optimal $(r,\delta)$-LRCs that attain the Singleton-type bound is an active and important research direction, particularly due to their practical applications in distributed storage systems. In this paper, we focus on the construction of optimal $(r,\delta)$-LRCs with flexible minimum distances, especially for the case $\delta \geq 3$. We first extend a general framework -- originally proposed by Li \textit{et al.} (IEEE Trans. Inf. Theory, vol. 65, no. 1, 2019) and Ma and Xing (J. Comb. Theory Ser. A., vol. 193, 2023) -- for constructing optimal $r$-LRCs via automorphism groups of elliptic function fields to the case of $(r,\delta)$-LRCs. This newly extended general framework relies on certain conditions concerning the group law of elliptic curves. By carefully selecting elliptic function fields suitable for this framework, we arrive at several families of explicit $q$-ary optimal $(r,3)$-LRCs and $(2,\delta)$-LRCs with lengths slightly less than $q + 2\sqrt{q}$. Next, by employing automorphism groups of hyperelliptic function fields of genus $2$, we develop a framework for constructing optimal $(r,3)$-LRCs and obtain a family of explicit $q$-ary optimal $(4,3)$-LRCs with code lengths slightly below $q+4\sqrt{q}$. We then consider the construction of optimal $(r,\delta)$-LRCs via hyperelliptic function fields of arbitrary genus $g \geq 2$, yielding a class of explicit $q$-ary optimal $(g+1-g',g+1+g')$-LRCs for $0 \leq g' \leq g-1$ with lengths up to $q + 2g\sqrt{q}$. Finally, applying certain superelliptic curves derived from modified Norm-Trace curves, we construct two families of explicit optimal $(r,\delta)$-LRCs with even longer code lengths and more flexible parameters. Notably, many of the newly constructed optimal $(r,\delta)$-LRCs attain the largest known lengths among existing constructions with flexible minimum distances.