Alphabet-Dependent Bounds for Linear Locally Repairable Codes Based on Residual Codes

TL;DR

This paper introduces an alphabet-dependent bound for linear locally repairable codes (LRCs) that accounts for the precise number of nodes needed for local repair, improving understanding of rate-distance tradeoffs in distributed storage.

Contribution

It proposes a new definition of locality, derives a novel alphabet-dependent bound based on residual codes, and extends existing bounds to better characterize LRC performance.

Findings

New bound applies to both initial and residual code definitions

Bound extends and improves upon Cadambe-Mazumdar and Singleton bounds

Provides the tightest known upper bound for large relative minimum distances

Abstract

Locally repairable codes (LRCs) have gained significant interest for the design of large distributed storage systems as they allow a small number of erased nodes to be recovered by accessing only a few others. Several works have thus been carried out to understand the optimal rate-distance tradeoff, but only recently the size of the alphabet has been taken into account. In this paper, a novel definition of locality is proposed to keep track of the precise number of nodes required for a local repair when the repair sets do not yield MDS codes. Then, a new alphabet-dependent bound is derived, which applies both to the new definition and the initial definition of locality. The new bound is based on consecutive residual codes and intrinsically uses the Griesmer bound. A special case of the bound yields both the extension of the Cadambe-Mazumdar bound and the Singleton-type bound for codes with locality $(r, {\delta})$, implying that the new bound is at least as good as these bounds. Furthermore, an upper bound on the asymptotic rate-distance tradeoff of LRCs is derived, and yields the tightest known upper bound for large relative minimum distances. Achievability results are also provided by deriving the locality of the family of Simplex codes together with a few examples of optimal codes.