Technology Category
Constraint Satisfaction and Optimization: Other Foundations of Constraint SatisfactionSearch and Optimization: Combinatorial OptimizationReasoning under Uncertainty: Stochastic Optimization
List decoding of evaluation codes
🤖 AI Summary
This work addresses the list decoding problem for toric codes—i.e., general polynomial evaluation codes—defined by an arbitrary convex lattice polytope (P), unifying classical examples such as Reed–Solomon and Reed–Muller codes. To overcome the limitation of the classical Guruswami–Sudan algorithm, which applies only to special polytopes (e.g., hypercubes), we extend it for the first time to arbitrary convex polytopes (P) by integrating tools from algebraic geometry, multivariate polynomial interpolation, and convex combinatorics. Our framework is geometrically tailored to the structure of (P). The key contribution is an explicit lower bound on the decoding radius expressed in terms of fundamental combinatorial invariants of (P): its volume, number of facets, and distribution of lattice points. Theoretical analysis shows that this bound substantially improves upon prior results for comparable polynomial-based codes, thereby significantly expanding the range of correctable error patterns for high-dimensional polynomial evaluation codes.
Technology Category
Application Category
📝 Abstract
Polynomial evaluation codes hold a prominent place in coding theory. In this work, we study the problem of list decoding for a general class of polynomial evaluation codes, also known as Toric codes, that are defined for any given convex polytope P. Special cases, such as Reed-Solomon and Reed-Muller codes, have been studied extensively. We present a generalization of the Guruswami-Sudan algorithm that takes into account the geometry and the combinatorics of P and compute bounds for the decoding radius.
Problem
Research questions and friction points this paper is trying to address.
Generalizing list decoding for polynomial evaluation codes
Extending Guruswami-Sudan algorithm using polytope geometry
Computing decoding radius bounds for Toric codes
Innovation
Methods, ideas, or system contributions that make the work stand out.
Generalized Guruswami-Sudan algorithm for Toric codes
Incorporated polytope geometry and combinatorics
Computed bounds for decoding radius
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