Optimal locally recoverable codes with hierarchy from nested $F$-adic expansions

TL;DR

This paper introduces a new method for constructing optimal hierarchical locally recoverable codes using algebraic number theory, enabling flexible parameters and improved code properties.

Contribution

It presents a novel algebraic construction of optimal hierarchical locally recoverable codes with adjustable parameters and fewer restrictions on the base field.

Findings

Codes with larger minimum distance and dimension

Flexible parameters for hierarchy size and field size

Optimality maintained across various configurations

Abstract

In this paper we construct new optimal hierarchical locally recoverable codes. Our construction is based on a combination of the ideas of \cite{ballentine2019codes,sasidharan2015codes} with an algebraic number theoretical approach that allows to give a finer tuning of the minimum distance of the intermediate code (allowing larger dimension of the final code), and to remove restrictions on the arithmetic properties of $q$ compared with the size of the locality sets in the hierarchy. In turn, we manage to obtain codes with a wide set of parameters both for the size $q$ of the base field, and for the hierarchy size, while keeping the optimality of the codes we construct.