Exact diameters of CD(k,q)

Prove that diam(CD(3,q))=6 for every prime power q, and that diam(CD(k,q))=k+5 for odd k>3 and k+4 for even k when q is sufficiently large.

Background

The paper gives computational diameter data and proves only selected cases of the proposed formula. The conjecture combines a uniform exact value for k=3 with asymptotically exact formulas for larger k.

References

Conjecture 8. ([115]) The diameter of CD(3, q) is 6 for all prime powers q. The diameter of CD(k, q) is k + 5, if k > 3 is odd, and k + 4, if k is even, provided that q is a large enough prime power.

— Some families of graphs, hypergraphs and digraphs defined by systems of equations  (2503.07915 - Lazebnik et al., 10 Mar 2025) in Conjecture 8, Section 6.5