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

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.