Let D(n) be the maximal determinant for nΓn{Β±1}-matrices, and R(n)=D(n)/nn/2 be the ratio of
D(n) to the Hadamard upper bound. Using the probabilistic method,
we prove new lower bounds on D(n) and R(n) in terms of
d=nβh, where h is the order of a Hadamard matrix and h is maximal
subject to hβ€n. For example, R(n)>(Οe/2)βd/2 if 1β€dβ€3, and R(n)>(Οe/2)βd/2(1βd2(Ο/(2h))1/2) if d>3. By a recent result of Livinskyi, d2/h1/2β0 as nββ,
so the second bound is close to (Οe/2)βd/2 for large n. Previous
lower bounds tended to zero as nββ with d fixed, except in the
cases dβ{0,1}. For dβ₯2, our bounds are better for all
sufficiently large n. If the Hadamard conjecture is true, then dβ€3, so
the first bound above shows that R(n) is bounded below by a positive
constant (Οe/2)β3/2>0.1133.Comment: 17 pages, 2 tables, 24 references. Shorter version of
arXiv:1402.6817v4. Typos corrected in v2 and v3, new Lemma 7 in v4, updated
references in v5, added Remark 2.8 and a reference in v6, updated references
in v