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Asymptotic count of Hadamard matrices

Ascertain the correct asymptotic order of the number of Hadamard matrices of order m. Specifically, determine whether the lower bound 2^{Ω(m log m)} gives the correct asymptotic scaling of the total number of Hadamard matrices (up to equivalence).

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Background

Counting Hadamard matrices as a function of order m remains a challenging combinatorial problem. Trivial lower bounds arise from row permutations, while upper bounds come from structural constraints; recent work has improved upper bounds.

The paper highlights this open problem and notes a conjecture that the lower bound 2{Ω(m log m)} is the correct asymptotic growth rate.

References

Another open problem of interest with Hadamard matrices involves the number of such possible matrices of a given size $m$. By considering only the $m!$ permutations of the rows, then a lower bound on the count of Hadamard matrices is $2{\Omega(m \log m)}$ while an upper bound of $2{\binom{m+1}2}$ is also easy to attain (see for an overview of this conjecture, along with the current improved upper bound of order $2{(1-c)m2/2}$ for a fixed constant $c$; it is further conjectured the lower bound is the correct asymptotic scaling).

— Complete pivoting growth of butterfly matrices and butterfly Hadamard matrices (2410.06477 - Peca-Medlin, 9 Oct 2024) in Section 4 (Butterfly Hadamard matrices)