Locally Repairable Codes and Matroid Theory

Bounds on the maximal minimum distance of linear locally repairable codes

“…Finally, the inequalities in (7) will be proved. A full version of the proof can be found in [15]. Lastly, we show that the bound in Thm.…”

“…Then we use Theorem 3.3 to show that it yields perfect matroids (and thus linear LRCs) for the desired class of parameters (n, k, r, δ), which are shown to equal their primed counterparts. The details of this can be found in [15].…”

“…4.2 for matroids (linear LRCs) from Construction 1 is tight for parameter sets (n, k, r, δ) for which there exists no perfect matroid (linear LRC) from Construction 1. Proof: A more detailed proof can be found in [15]. Let M = M (F 1 , ..., F m ; k; ρ) be a matroid from Construction 1 for which there exists no perfect matroid from the same construction with the same parameters (n, k, r, δ).…”

“…Secondly, we improve the general lower bound on d max (n, k, r, δ) for linear LRCs and prove that the new bound is optimal for the matroid-based construction. The results of this paper were originally presented in the bachelor thesis of the first author [15], which provides a more comprehensive account as well as full proofs.…”