# Combinatorial Alphabet-Dependent Bounds for Locally Recoverable Codes

## TL;DR

This paper introduces new combinatorial and linear programming bounds for locally recoverable codes, improving the understanding of their rate-distance trade-offs especially over small alphabets.

## Contribution

It presents novel combinatorial bounds and an LP-based approach that yield tighter estimates on the rate of LRC codes with specified distance.

## Key findings

- New combinatorial bounds including sphere packing and Plotkin bounds.
- An LP bound that outperforms existing bounds in examples.
- The tightest known upper bound on the rate of linear LRC codes with given distance.

## Abstract

Locally recoverable (LRC) codes have recently been a focus point of research in coding theory due to their theoretical appeal and applications in distributed storage systems. In an LRC code, any erased symbol of a codeword can be recovered by accessing only a small number of other symbols. For LRC codes over a small alphabet (such as binary), the optimal rate-distance trade-off is unknown. We present several new combinatorial bounds on LRC codes including the locality-aware sphere packing and Plotkin bounds. We also develop an approach to linear programming (LP) bounds on LRC codes. The resulting LP bound gives better estimates in examples than the other upper bounds known in the literature. Further, we provide the tightest known upper bound on the rate of linear LRC codes with a given relative distance, an improvement over the previous best known bounds.