Large-determinant sign matrices of order 4k+1

Abstract

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.