Decoding Downset codes over a finite grid

TL;DR

This paper extends a deterministic decoding algorithm originally designed for bounded degree polynomials to a broader class of Downset codes, which are defined by monomials closed under factors, over finite grids.

Contribution

It adapts existing decoding techniques to the general family of Downset codes, broadening their applicability in coding theory.

Findings

Decoding algorithm successfully adapted for Downset codes

Efficient decoding over general grids demonstrated

Broader class of codes can now be decoded reliably

Abstract

In a recent paper, Kim and Kopparty (Theory of Computing, 2017) gave a deterministic algorithm for the unique decoding problem for polynomials of bounded total degree over a general grid. We show that their algorithm can be adapted to solve the unique decoding problem for the general family of Downset codes. Here, a downset code is specified by a family D of monomials closed under taking factors: the corresponding code is the space of evaluations of all polynomials that can be written as linear combinations of monomials from D.