Optimal Quaternary Locally Repairable Codes Attaining the Singleton-like Bound

TL;DR

This paper investigates optimal quaternary locally repairable codes that meet the Singleton-like bound, providing new necessary conditions and explicit constructions for 27 classes of such codes using finite geometry tools.

Contribution

It introduces a parity-check matrix approach to characterize optimal quaternary LRCs and identifies 27 parameter classes with explicit constructions.

Findings

27 classes of optimal quaternary LRCs identified

Necessary conditions for optimality established

Explicit constructions provided for each class

Abstract

Recent years, several new types of codes were introduced to provide fault-tolerance and guarantee system reliability in distributed storage systems, among which locally repairable codes (LRCs for short) have played an important role. A linear code is said to have locality $r$ if each of its code symbols can be repaired by accessing at most $r$ other code symbols. For an LRC with length $n$, dimension $k$ and locality $r$, its minimum distance $d$ was proved to satisfy the Singleton-like bound $d\leq n-k-\lceil k/r\rceil+2$. Since then, many works have been done for constructing LRCs meeting the Singleton-like bound over small fields. In this paper, we study quaternary LRCs meeting Singleton-like bound through a parity-check matrix approach. Using tools from finite geometry, we provide some new necessary conditions for LRCs being optimal. From this, we prove that there are $27$ different classes of parameters for optimal quaternary LRCs. Moreover, for each class, explicit constructions of corresponding optimal quaternary LRCs are presented.