Diameter bound for CD(k,q)

Prove that there exists a positive constant C such that diam(CD(k,q))≤(log_{q−1}q)k+C for all k≥2 and all prime powers q.

Background

The paper studies the diameters of the connected components CD(k,q) and reports computational data and several exact values. The conjecture proposes a uniform linear-in-k upper bound with coefficient log_{q−1}q.

References

Conjecture 7. ([82]) There exists a positive constant C such that for all k ≥ 2, and all prime powers q, diam(CD(k, q)) ≤ (logq−1 q)k + C.

— Some families of graphs, hypergraphs and digraphs defined by systems of equations  (2503.07915 - Lazebnik et al., 10 Mar 2025) in Conjecture 7, Section 6.5