Preprint

Inserted: 17 apr 2026

Year: 2026

Abstract:

We derive the asymptotic formula $\alpha(k,q)=\lambda_{k-1}q^k+o(q^k)$, where $\alpha(k,q)$ is the independence number of the de Bruijn graph $B(k,q)$, and $\lambda_{k-1}$ is a constant arising from a variational problem on the unit $(k-1)$-dimensional cube. When $k=4$, we show the bounds $91/240\le \lambda_3\le 11/28$. For odd prime $k$, we analyse the binary case $q=2$ via a phase reduction on rotation orbits. For $k=11$ and $k=13$ this yields certified optimal constructions, which combined with a lifting theorem by Lichiardopol gives exact formulas for $\alpha(11,q)$ and $\alpha(13,q)$ for all $q\ge2$, extending the known cases $k=3,5,7$.