There are no good infinite families of toric codes

TL;DR

This paper proves that good infinite families of toric codes do not exist by establishing a Szemerédi-type result about the unavoidable presence of large hypercubes in dense subsets of high-dimensional grids.

Contribution

It introduces a general Szemerédi-type theorem that implies the non-existence of good infinite families of toric codes.

Findings

Good infinite families of toric codes do not exist.

Dense subsets in high-dimensional grids contain arbitrarily large hypercubes.

The result generalizes Szemerédi's theorem to hypercubes in multidimensional settings.

Abstract

Soprunov and Soprunova introduced the notion of a good infinite family of toric codes. We prove that such good families do not exist by proving a more general Szemer\'edi-type result: for all $c\in(0,1]$ and all positive integers $N$, subsets of density at least $c$ in $\{0,1,\dots,N-1\}^n$ contain hypercubes of arbitrarily large dimension as $n$ grows.