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

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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−⌈kr⌉+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.