Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

There are no good infinite families of toric codes

Jason P. Bell, Sean Monahan, Matthew Satriano, Karen Situ, Zheng Xie

TL;DR

The paper proves there are no good infinite families of toric codes by establishing a Szemerédi-type density phenomenon for dense subsets of high-dimensional grids: dense sets contain arbitrarily large hypercubes under injective integer affine maps. It develops a lower bound showing $f_N(n,c)$ grows at least like $\log n$, and a complementary upper-bound construction using the Lovász Local Lemma to demonstrate the existence of dense sets with small embedding dimension, thereby limiting the potential for large information rate and minimum distance simultaneously. Consequently, either the information rate $R(P)$ or the relative minimum distance $d(P)$ must degenerate along any infinite family, ruling out goodness. The results extend to an entropy-parameterized view and solidify the conjecture that there are no good infinite families of toric codes, with implications for code design via toric varieties and connections to additive combinatorics.

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 Szemerédi-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.

There are no good infinite families of toric codes

TL;DR

The paper proves there are no good infinite families of toric codes by establishing a Szemerédi-type density phenomenon for dense subsets of high-dimensional grids: dense sets contain arbitrarily large hypercubes under injective integer affine maps. It develops a lower bound showing grows at least like , and a complementary upper-bound construction using the Lovász Local Lemma to demonstrate the existence of dense sets with small embedding dimension, thereby limiting the potential for large information rate and minimum distance simultaneously. Consequently, either the information rate or the relative minimum distance must degenerate along any infinite family, ruling out goodness. The results extend to an entropy-parameterized view and solidify the conjecture that there are no good infinite families of toric codes, with implications for code design via toric varieties and connections to additive combinatorics.

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 Szemerédi-type result: for all and all positive integers , subsets of density at least in contain hypercubes of arbitrarily large dimension as grows.
Paper Structure (3 sections, 10 theorems, 38 equations)

This paper contains 3 sections, 10 theorems, 38 equations.

Table of Contents

  1. Introduction
  2. Lower bound for the growth rate of $f_N$
  3. Proof of the main theorems

Key Result

theorem 1.5

conj:main is true. Therefore, there are no good infinite families of toric codes (in the sense of def:good family of toric codes).

Theorems & Definitions (26)

  • definition 1.1: dolorfino2022good-families-toric
  • definition 1.2: soprunov-soprunova2010bringing-toric-codes, cf. dolorfino2022good-families-toric
  • definition 1.3: dolorfino2022good-families-toric
  • conjecture 1.4: dolorfino2022good-families-toric
  • theorem 1.5
  • definition 1.6
  • remark 1.7
  • proposition 1.8
  • theorem 1.9
  • remark 1.10
  • ...and 16 more