Bounds and Constructions for Linear Locally Repairable Codes over Binary Fields

摘要

For binary $[n,k,d]$ linear locally repairable codes (LRCs), two new upperbounds on $k$ are derived. The first one applies to LRCs with disjoint localrepair groups, for general values of $n,d$ and locality $r$, containing somepreviously known bounds as special cases. The second one is based on solving anoptimization problem and applies to LRCs with arbitrary structure of localrepair groups. Particularly, an explicit bound is derived from the second boundwhen $d\geq 5$. A specific comparison shows this explicit bound outperforms theCadambe-Mazumdar bound for $5\leq d\leq 8$ and large values of $n$. Moreover, aconstruction of binary linear LRCs with $d\geq6$ attaining our second bound isprovided.