Asymptotic construction of locally repairable codes with multiple recovering sets

TL;DR

This work addresses the problem of constructing asymptotically good locally repairable codes (LRCs) with multiple recovering sets (availability) by leveraging automorphism groups of Garcia-Stichtenoth towers. It develops a group-theoretic framework that yields two disjoint recovering sets via fixed-field subextensions, and then applies two Garcia-Stichtenoth tower variants to derive explicit asymptotic rate–distance tradeoffs. The main contributions are two asymptotic constructions (I and II) that produce $q$-ary $[n,k,d; (r_1,r_2)]$-LRCs with provable bounds on $\delta$ and $R$ depending on tower parameters and locality choices, providing greater locality flexibility than prior work. These results advance scalable, distributed-storage-friendly codes by offering new parameter regimes for LRCs with multiple recovering sets and establishing their asymptotic viability using function-field automorphisms.

Abstract

Locally repairable codes have been extensively investigated due to practical applications in distributed and cloud storage systems in recent years. However, not much work on asymptotic behavior of locally repairable codes has been done. In particular, there is few result on constructive lower bound of asymptotic behavior of locally repairable codes with multiple recovering sets. In this paper, we construct some families of asymptotically good locally repairable codes with multiple recovering sets via automorphism groups of function fields of the Garcia-Stichtenoth towers. The main advantage of our construction is to allow more flexibility of localities.

Theorems & Definitions (13)

• Definition 1
• Proposition 2.1
• Proposition 2.2
• proof
• Proposition 3.1
• proof
• Lemma 3.2
• proof
• Theorem 3.3
• proof