Universal lower bound for norm-function parameters

Determine whether the inequality $b_q^d>q$ holds for every prime power $q\ge 3$ and every integer $d\ge 3$, where $b_q^d=|\{x\in F_{q^d}:N_q^d(x-1)=N_q^d(x)-1\}|$ and $N_q^d:F_{q^d}\to F_q$ is the norm map.

Background

The paper studies Neumaier graphs constructed from the norm map of the finite-field extension Fqd/FqF_{q^d}/F_q. Theorem \ref{thm:norm} shows that the construction yields a strictly Neumaier graph whenever bqd>qb_q^d>q, while Theorem \ref{thm:norm infinite} proves this inequality for infinitely many parameter choices, including all odd d≥3d\ge 3 when qq is sufficiently large.

The authors note that computed values suggest the inequality may hold under substantially weaker hypotheses, namely for every prime power q≥3q\ge 3 and every integer d≥3d\ge 3. Establishing this would substantially broaden the range for which the norm construction is known to produce strictly Neumaier graphs. They further mention a Lang--Weil estimate that lowers the sufficient threshold on qq, but does not resolve the stated universal range.

References

Perhaps this is true as soon as $q\ge 3$ and $d\ge 3$.

— A unified framework for existing and new constructions of Neumaier graphs  (2610.01273 - Abiad et al., 1 Oct 2026) in Section 5, immediately following the proof of Theorem 5.2 (Theorem \ref{thm:norm infinite}), before Example \ref{exa:d=2}