Technology Category
Machine Learning: Matrix & Tensor MethodsKnowledge Representation and Reasoning: Computational Complexity of ReasoningConstraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction
Duals of multiplicity codes
🤖 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
Application Category
📝 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
Research questions and friction points this paper is trying to address.
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
Methods, ideas, or system contributions that make the work stand out.
Uses Gröbner basis for dimension computation
Describes dual via indicator functions
Provides explicit parity-check matrix
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