Optimal 5-Seq LRCs With Availability From Golomb Rulers

Abstract

In this paper, we propose a simple construction for binary $(n,k)$ linear codes using s-mark Golomb rulers. We prove that these codes are sequential-recovery locally repairable codes (LRCs) with availability 2, which can sequentially recover 5 erased symbols. We prove the necessary and sufficient condition for the proposed codes to be rate-optimal. We also prove the necessary and sufficient condition for the proposed codes to be dimension-optimal. Finally, we propose some variations of this constructions to obtain some 5-sequential recovery LRCs with availability 3. The proposed codes have higher rates and have more flexible choice for the lengths than other previously reported constructions.