Duals of multiplicity codes
Eduardo Camps Moreno, Adrián Fidalgo-Díaz, Hiram H. López, Umberto Martínez-Peñas, Diego Ruano, Rodrigo San-José
TL;DR
This work analyzes the duals of multivariate multiplicity codes built from Hasse derivatives on Cartesian grids, combining Gröbner basis methods and Hermite interpolation to obtain explicit dual descriptions and parity-check matrices. It establishes that duality is not closed within the family for $m>1$ and $r>1$, in contrast to univariate or $r=1$ cases, and uses the dual constructions to derive a lower bound on the minimum folded distance of duals. The authors develop a framework based on isometries for the folded metric, एकnd construct Hermite interpolation bases that underpin the dual representations, providing concrete bases and generators for duals in general and for key special cases. These results yield practical syndrome-based decoding tools for multivariate multiplicity codes and extend to general multiplicity codes via footprint-based dimension formulas and dual bases, enabling parity-check matrix constructions in wide settings.
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.