Optimal $(2,\delta)$ Locally Repairable Codes via Punctured Simplex Codes

TL;DR

This paper introduces new constructions of optimal $(2, oldsymbol{\delta})$-locally repairable codes (LRCs) over finite fields, using finite geometry and character sums, achieving optimality and new parameter sets.

Contribution

It provides a novel condition for punctured simplex codes to be $(2, oldsymbol{\delta})$-LRCs and derives multiple infinite families with optimal parameters.

Findings

All new LRCs meet the generalized Cadambe-Mazumdar bound.

Some codes are Griesmer or distance-optimal.

The constructions are flexible and extend previous results.

Abstract

Locally repairable codes (LRCs) have attracted a lot of attention due to their applications in distributed storage systems. In this paper, we provide new constructions of optimal $(2, \delta)$-LRCs over $\mathbb{F}_q$ with flexible parameters. Firstly, employing techniques from finite geometry, we introduce a simple yet useful condition to ensure that a punctured simplex code becomes a $(2, \delta)$-LRC. It is worth noting that this condition only imposes a requirement on the size of the puncturing set. Secondly, utilizing character sums over finite fields and Krawtchouk polynomials, we determine the parameters of more punctured simplex codes with puncturing sets of new structures. Several infinite families of LRCs with new parameters are derived. All of our new LRCs are optimal with respect to the generalized Cadambe-Mazumdar bound and some of them are also Griesmer codes or distance-optimal codes.