There are no good infinite families of toric codes
🤖 AI Summary
This work resolves a fundamental question on the asymptotic goodness of toric codes: it proves that no asymptotically good infinite family of toric codes exists. To achieve this, the authors establish a novel high-dimensional Szemerédi-type density theorem—the first systematic application of combinatorial density methods to algebraic coding theory. The theorem asserts that, for any density $c > 0$ and dimension $n o infty$, every subset of the grid ${0,dots,N-1}^n$ with density at least $c$ must contain arbitrarily large hypercube configurations. This result rigorously refutes the Soprunov–Soprunova conjecture asserting the existence of an asymptotically good infinite family of toric codes, thereby exposing an intrinsic performance bottleneck of toric codes in the asymptotic regime. The proof integrates tools from combinatorial number theory, discrete geometry, and lattice density analysis, establishing a new paradigm for investigating structural limits in algebraic coding theory.
Application Category
📝 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 $cin(0,1]$ 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.
Problem
Research questions and friction points this paper is trying to address.
toric codes
infinite scalability
mathematical principles
Innovation
Methods, ideas, or system contributions that make the work stand out.
Torric Codes
Infinite Series
Hypercube Combinatorics
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