Optimal Quantum $(r,\delta)$-Locally Repairable Codes From Matrix-Product Codes

TL;DR

This paper develops conditions for constructing optimal quantum locally repairable codes using matrix-product codes, introducing new families with flexible parameters for improved quantum error correction.

Contribution

It provides a necessary and sufficient condition for MP codes to be optimal quantum LRCs and introduces five infinite families with flexible parameters.

Findings

Established criteria for optimal quantum $(r,oldsymbol{ heta})$-LRCs from MP codes

Characterized optimal quantum LRCs from nested and non-nested MP codes

Presented five infinite families of optimal quantum LRCs with flexible parameters

Abstract

This paper studies optimal quantum $(r,\delta)$-LRCs from matrix-product (MP) codes. We establish a necessary and sufficient condition for an MP code to be an optimal $(r,\delta)$-LRC. Based on this, we present a characterization for optimal quantum $(r,\delta)$-LRCs from MP codes with nested constituent codes, and also study optimal quantum $(r,\delta)$-LRCs constructed from MP codes with non-nested constituent codes. Through Hermitian dual-containing and Euclidean dual-containing MP codes, we present five infinite families of optimal quantum $(r,\delta)$-LRCs with flexible parameters.