Optimal Locally Recoverable Codes With Hierarchy From Nested F-Adic Expansions

In this paper we construct new optimal hierarchical locally recoverable codes. Our construction is based on a combination of the ideas of Ballentine et al., (2019) and Sasidharan et al., (2015) 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 wider 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.