Minimum Distance and Parameter Ranges of Locally Recoverable Codes with Availability from Fiber Products of Curves

TL;DR

This paper constructs and analyzes families of locally recoverable codes with availability using fiber products of curves, determining their minimum distance and exploring their parameters to show they can achieve high rates and low relative defect.

Contribution

It introduces a new fiber product construction for locally recoverable codes with availability, providing exact minimum distance formulas and parameter analysis.

Findings

Fiber product codes can achieve arbitrarily large rate.

They can have arbitrarily small relative defect.

The paper provides a general minimum distance theorem.

Abstract

We construct families of locally recoverable codes with availability $t\geq 2$ using fiber products of curves, determine the exact minimum distance of many families, and prove a general theorem for minimum distance of such codes. The paper concludes with an exploration of parameters of codes from these families and the fiber product construction more generally. We show that fiber product codes can achieve arbitrarily large rate and arbitrarily small relative defect, and compare to known bounds and important constructions from the literature.