We derive the asymptotic formula $α(k,q)=λ_{k-1}q^k+o(q^k)$, where $α(k,q)$ is the independence number of the de Bruijn graph $B(k,q)$, and $λ_{k-1}$ is a constant arising from a variational problem on the unit $(k-1)$-dimensional cube. When $k=4$, we show the bounds $91/240\le λ_3\le 11/28$. For odd prime $k$, we analyse the binary case $q=2$ via a phase reduction on rotation orbits. For $k=11,13,17$ this yields compact orbit-marker certificates for optimal constructions. Combined with a lifting theorem by Lichiardopol, these certificates give exact formulas for $α(11,q)$, $α(13,q)$, and $α(17,q)$ for all $q\ge2$, extending the known cases $k=3,5,7$.

翻译：我们推导出渐近公式$α(k,q)=λ_{k-1}q^k+o(q^k)$，其中$α(k,q)$是de Bruijn图$B(k,q)$的独立数，而$λ_{k-1}$是源于单位$(k-1)$维立方体上变分问题的常数。当$k=4$时，我们证明了上界$91/240\le λ_3\le 11/28$。对于奇数素数$k$，我们通过旋转轨道的相位约化分析二元情形$q=2$。对于$k=11,13,17$，这给出了最优构造的紧凑轨道标记证书。结合Lichiardopol的提升定理，这些证书对所有$q\ge2$给出了$α(11,q)$、$α(13,q)$和$α(17,q)$的精确公式，扩展了已知情形$k=3,5,7$。