# Bounds and Constructions of Codes with All-Symbol Locality and Availability

## TL;DR

This paper derives new bounds on the minimum distance of all-symbol locally recoverable codes with availability and proposes explicit constructions using rank-metric codes, expander graphs, and existing LRC codes.

## Contribution

It introduces improved upper bounds on code distance and provides novel explicit constructions that work across different rate regions without restricting alphabet size.

## Key findings

- New upper bounds on minimum distance of LRC codes with availability.
- Explicit constructions using expander graphs and rank-metric codes.
- The constructions outperform existing methods in specific rate regions.

## Abstract

We investigate the distance properties of linear locally recoverable codes (LRC codes) with all-symbol locality and availability. New upper and lower bounds on the minimum distance of such codes are derived. The upper bound is based on the shortening method and improves existing shortening bounds. To reduce the gap in between upper and lower bounds we do not restrict the alphabet size and propose explicit constructions of codes with locality and availability via rank-metric codes. The first construction relies on expander graphs and is better in low rate region, the second construction utilizes LRC codes developed by Wang et al. as inner codes and better in high rate region.