On Good Infinite Families of Toric Codes or the Lack Thereof

TL;DR

This work analyzes the existence of good infinite families of toric codes defined by lattice polytopes, focusing on how polytope operations like join and direct sum affect the key parameters $\delta$ and $R$. It derives explicit formulas for $\delta$ and $R$ under these constructions and demonstrates that several natural infinite-family constructions (boxes, simplices, and iterative joins) fail to be good, providing strong evidence against the existence of any good infinite family. The authors connect these obstructions to unit hypercubes and Minkowski length, showing that unbounded hypercube content or unbounded Minkowski length forces $\delta\to0$, and conjecture that bounded Minkowski length forces $R\to0$; they verify this in several special cases. The paper thus argues, with substantial evidence, that no good infinite family of toric codes exists and outlines precise avenues for proving or refuting the central conjecture, with implications for the design of high-dimensional toric-code families.

Abstract

A toric code, introduced by Hansen to extend the Reed-Solomon code as a $k$-dimensional subspace of $\mathbb{F}_q^n$, is determined by a toric variety or its associated integral convex polytope $P \subseteq [0,q-2]^n$, where $k=|P \cap \mathbb{Z}^n|$ (the number of integer lattice points of $P$). There are two relevant parameters that determine the quality of a code: the information rate, which measures how much information is contained in a single bit of each codeword; and the relative minimum distance, which measures how many errors can be corrected relative to how many bits each codeword has. Soprunov and Soprunova defined a good infinite family of codes to be a sequence of codes of unbounded polytope dimension such that neither the corresponding information rates nor relative minimum distances go to 0 in the limit. We examine different ways of constructing families of codes by considering polytope operations such as the join and direct sum. In doing so, we give conditions under which no good family can exist and strong evidence that there is no such good family of codes.

Figures (7)

• Figure 1: The polytope $P$.
• Figure 2: (a) Polytope $P$ (b) Polytope $Q$
• Figure 3: The Polytope $P * Q$
• Figure 4: The polytope $P$.
• Figure 5: (a) Polytope $P_1$ (b) Polytope $P_2$ (c) Polytope $P_3$ (d) Polytope $P_4=P_5$

Theorems & Definitions (61)

• Theorem 1.1
• Example 2.1
• Definition 2.2
• Definition 2.3
• Definition 2.4
• Remark 2.5
• Remark 2.6
• Definition 2.7
• Definition 2.8
• Proposition 2.9