On the Asymptotic Order of Circuit Codes

TL;DR

This paper establishes the asymptotic maximum length of $d$-dimensional circuit codes of a given spread $k$, showing it grows exponentially with $d$ with a specific logarithmic correction.

Contribution

It provides a precise asymptotic characterization of the maximum length of circuit codes in high dimensions, improving understanding of their growth rate.

Findings

Maximum length of circuit codes is $2^{d+O_k( ext{log}^2 d)}$

Growth rate is exponential in dimension $d$

Constant depends only on spread $k$

Abstract

In this note we prove that the maximum length of a $d$-dimensional circuit code of spread $k$ equals $2^{d+O_k(\log^2d)}$, with the implied constant depending only on $k$.