On Finding the Largest Minimum Distance of Locally Recoverable Codes
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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.
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