Duals of multiplicity codes

TL;DR

This paper investigates the duals of multivariate multiplicity codes, providing explicit descriptions, dimension calculations, and bounds on minimum distance, revealing that their duals are not equivalent to the original codes unlike other evaluation codes.

Contribution

It offers the first explicit characterization of duals of multivariate multiplicity codes, including their dimension, dual functions, and parity-check matrices, highlighting their non-isometric nature.

Findings

Duals are not equivalent or isometric to original codes.

Explicit parity-check matrices are constructed.

Lower bounds on dual code minimum distance are established.

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.