On Finding the Largest Minimum Distance of Locally Recoverable Codes

TL;DR

This paper investigates the maximum minimum distance of locally recoverable codes, converts the problem into graph theory, and establishes connections to extremal graph theory, providing new insights and extensions.

Contribution

It introduces a graph-theoretic formulation of the largest minimum distance problem for LRCs and links it to extremal graph theory, enabling new derivations and understanding.

Findings

Converted LMD to a graph theory problem

Proved the equivalence between LMD and the graph problem

Connected LMD to open problems in extremal graph theory

Abstract

The (n, k, r)-Locally recoverable codes (LRC) studied in this work are (n, k) linear codes for which the value of each coordinate can be recovered by a linear combination of at most r other coordinates. In this paper, we are interested to find the largest possible minimum distance of (n,k,r)-LRCs, denoted D(n,k,r). We refer to the problem of finding the value of D(n,k,r) as the largest minimum distance (LMD) problem. LMD can be approximated within an additive term of one; it is known in the literature that D(n,k,r) is either equal to d* or d*-1, where d*=n-k-ceil(k/r) +2. Also, in the literature, LMD has been solved for some ranges of code parameters n, k and r. However, LMD is still unsolved for the general code parameters. In this work, we convert LMD to a simply stated problem in graph theory, and prove that the two problems are equivalent. In fact, we show that solving the derived graph theory problem not only solves LMD, but also directly translates to construction of optimal LRCs. Using these new results, we show how to easily derive the existing results on LMD and extend them. Furthermore, we show a close connection between LMD and a challenging open problem in extremal graph theory; an indication that LMD is perhaps difficult to solve for general code parameters.