Duals of multiplicity codes
Information Theory
2025-05-21 v1 math.IT
Abstract
Multivariate multiplicity codes have been recently explored because of their importance for list decoding and local decoding. Given a multivariate multiplicity code, in this paper, we compute its dimension using Gr\"obner basis tools, its dual in terms of indicator functions, and explicitly describe a parity-check matrix. In contrast with Reed--Muller, Reed--Solomon, univariate multiplicity, and other evaluation codes, the dual of a multivariate multiplicity code is not equivalent or isometric to a multiplicity code (i.e., this code family is not closed under duality). We use our explicit description to provide a lower bound on the minimum distance for the dual of a multiplicity code.
Cite
@article{arxiv.2505.14472,
title = {Duals of multiplicity codes},
author = {Eduardo Camps Moreno and Adrián Fidalgo-Díaz and Hiram H. López and Umberto Martínez-Peñas and Diego Ruano and Rodrigo San-José},
journal= {arXiv preprint arXiv:2505.14472},
year = {2025}
}