Duals of multiplicity codes

2025-05-20

AI Summary

This work addresses the dimension computation, dual structure characterization, and parity-check matrix construction for multivariate multiplicity codes. A central challenge is that their duals are not closed under the multiplicity code family. To resolve this, we develop a unified analytical framework integrating Gröbner bases, algebraic coding theory, and polynomial interpolation. Our contributions are threefold: (i) we derive the first explicit closed-form formula for the dimension; (ii) we characterize the dual code via an indicator function on monomial supports and construct an explicit, efficiently computable parity-check matrix; and (iii) we rigorously prove that the dual of any nondegenerate multivariate multiplicity code is never equivalent to any multiplicity code—contradicting the duality intuition from classical families such as Reed–Muller codes. Furthermore, we establish a tight lower bound on the minimum distance of the dual code and present the first general dual description applicable to arbitrary dimension and multiplicity.

Technology Category

• Machine Learning: Matrix & Tensor Methods
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Security and Privacy: Applications of cryptography
• Web Mining and Content Analysis: Models for Web evolution

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.

Problem

• Computes dimension of multivariate multiplicity codes using Gröbner basis
• Describes dual codes via indicator functions and parity-check matrices
• Proves duality does not preserve code family equivalence or isometry

Innovation

• Uses Gröbner basis for dimension computation
• Describes dual via indicator functions
• Provides explicit parity-check matrix