List decoding of evaluation codes
Silouanos Brazitikos, Theodoulos Garefalakis, Eleni Tzanaki
TL;DR
The paper studies list decoding of polytope-based polynomial evaluation codes ${\mathcal{C}}_{P}$ (toric codes) defined by a lattice polytope $P$ over ${\mathbb F}_q$, generalizing Guruswami-Sudan to exploit the geometry of $P$. It develops a geometry-aware decoding framework that combines Newton polytopes, Minkowski sums, and Ehrhart theory to bound the decoding radius, using the pyramid relation $L_{\mathrm{Pyr}(P)}(\lambda)=\sum_{k=1}^{\lambda} L_P(k)$ to connect lattice-point counts with decoding performance. The basic method builds a bivariate auxiliary polynomial $Q(X,Y)$ and reduces decoding to root-finding of $Q(X,f(X))$, while the improved method imposes multiplicity-$r$ zeros at evaluation points to sharpen the radius; both are implemented with a linear-algebra core of complexity $O(|I|^3)$ and use $I=\lambda\mathrm{Pyr}(P)$ to tie parameters to Ehrhart counts. As an application, Reed-Muller codes arise as a special case (simplex $P$), yielding explicit radius bounds such as $s^m\bigl(1 - \frac{e^{3/2}}{m}(d/s)^{1/(m+1)}\bigr)^m$, illustrating the practical impact for multivariate codes.
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.