Large-determinant sign matrices of order 4k+14k+1

The Hadamard maximal determinant problem asks for the largest n×nn×n determinant with entries ±1±1. When n≡1(mod4), the maximal excess construction of Farmakis and Kounias [The excess of Hadamard matrices and optimal designs, Discrete Math. 67 (1987) 165–176] produces many large (though seldom maximal) determinants. For certain small nn, still larger determinants have been known; e.g., see [W.P. Orrick, B. Solomon, R. Dowdeswell, W.D. Smith, New lower bounds for the maximal determinant problem, arXiv preprint math.CO/0304410]. Here, we define “3-normalized” n×nn×n Hadamard matrices, and construct large (n+1)×(n+1)(n+1)×(n+1) determinants from them. Our constructions give most of the previous “small n  ” records, and set new records when n=37,49,65,73,77,85,89,93,97, and 101, most of which exceed the reach of the maximal excess technique. We suspect that our n=37n=37 determinant, 72×917×236 is best possible.