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arXiv · 2505.14472

Duals of multiplicity codes

Eduardo Camps Moreno·Adrián Fidalgo-Díaz·Hiram H. López·Umberto Martínez-Peñas·Diego Ruano·Rodrigo San-José
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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.

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Eduardo Camps Moreno, Adrián Fidalgo-Díaz, Hiram H. López, Umberto Martínez-Peñas, Diego Ruano, Rodrigo San-José. 2025-05-20. Duals of multiplicity codes. https://arxiv.org/abs/2505.14472

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