Duals of multiplicity codes
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 Gröbner 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.
Authors
- Hiram H. López (ORCID: https://orcid.org/0000-0002-9832-7145)
- Umberto Martínez-Peñas (ORCID: https://orcid.org/0000-0003-2626-8139)
- Rodrigo San–José (ORCID: https://orcid.org/0000-0002-0944-7584)
- Eduardo Camps-Moreno (ORCID: https://orcid.org/0000-0001-9662-8051)
- Diego Ruano (ORCID: https://orcid.org/0000-0001-7304-0087)
- Adrián Fidalgo-Díaz
Institutions
- Universidad de Valladolid (ES)
- Institut de Mathématiques de Bordeaux (FR)
- Virginia Tech (US)
Publication Details
- Journal
- Designs Codes and Cryptography
- Published
- 2026-04-01
- DOI
- https://doi.org/10.1007/s10623-026-01812-2
- Primary Topic
- Coding theory and cryptography
- Type
- article
- Field-Weighted Citation Impact
- 0.00
Funders
- National Science Foundation
- Universidad de Valladolid
- Junta de Castilla y León
- Simons Institute for the Theory of Computing, University of California Berkeley
- European Regional Development Fund
- Agencia Estatal de Investigación
- Ministerio de Universidades