Existence of true Hadamards beyond the verified range

Determine whether any k>7 admits a true Hadamard matrix in the half-flip family at length L=8k+3, or prove that the family contains true Hadamards only for k≤7.

Background

Exhaustive computation finds true Hadamards in the half-flip family for k∈{1,2,3,6,7} and none for the tested values 4,5,8,…,14. These results refute the earlier H4 periodicity hypothesis but do not establish the eventual behavior of the family.

The paper notes that extending the computation requires handling increasingly difficult quadratic constraints in the mod-16 stage.

References

Does any k > 7 admit a true Hadamard in the half-flip family, or is the family bounded by k \le 7?

— An explicit half-flip family of 32-modular Hadamard matrices at L = 3 mod 8: structural placement and mod-tower analysis  (2609.25543 - Kulhandjian, 22 Sep 2026) in Section 6, subsection “Falsification of the H4 hypothesis,” Open questions

Can these quadratic forms be decomposed into low-rank components, extending the polynomial-time verifier beyond $L \approx 100$?

— An explicit half-flip family of 32-modular Hadamard matrices at L = 3 mod 8: structural placement and mod-tower analysis  (2609.25543 - Kulhandjian, 22 Sep 2026) in Section 6, Open questions