  The Hadamard maximal determinant problem asks for the largest n-by-n
determinant with entries in {+1,-1}. When n is congruent to 1 (mod 4), the
maximal excess construction of Farmakis & Kounias has been the most successful
general method for constructing large (though seldom maximal) determinants. For
certain small n, however, still larger determinants have been known; several
new records were recently reported in ArXiv preprint math.CO/0304410 . Here, we
define ``3-normalized'' n-by-n Hadamard matrices, and construct
large-determinant matrices of order n+1 from them. Our constructions account
for most of the previous ``small n'' records, and set new records when n=37,
49, 65, 73, 77, 85, 93, and 97, most of which are beyond the reach of the
maximal excess technique. We conjecture that our n=37 determinant, 72 x 9^{17}
x 2^{36}, achieves the global maximum.
