On finding the largest minimum distance of locally recoverable codes: A graph theory approach

The $[n,k,r]$-Locally recoverable codes (LRC) studied in this work are a well-studied family of $[n,k]$ linear codes for which the value of each symbol can be recovered by a linear combination of at most r other symbols. In this paper, we study the LMD problem, which is to find the largest possible minimum distance of $[n,k,r]$-LRCs, denoted by $D(n,k,r)$. LMD can be approximated within an additive term of one—it is known that $D(n,k,r)$ is equal to either $d^{⁎}$ or $d^{⁎}-1$, where $d^{⁎}=n-k-\lceil \frac{k}{r}\rceil +2$. Moreover, for a range of parameters, it is known precisely whether the distance $D(n,k,r)$ is $d^{⁎}$ or $d^{⁎}-1$. However, the problem is still open despite a significant effort. In this work, we convert LMD to an equivalent simply-stated problem in graph theory. Using this conversion, we show that an instance of LMD is at least as hard as computing the size of a maximal graph of high girth, a hard problem in extremal graph theory. This is an evidence that LMD—although can be approximated within an additive term of one—is hard to solve in general. As a positive result, thanks to the conversion and the exiting results in extremal graph theory, we solve LMD for a range of code parameters that has not been solved before.