The maximum determinant of ± 1 matrices

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We give a new proof for the bound on the value of the determinant of a ± 1 matrix of dimension n ≡ 1 (mod 4) first given by Barba. Adapting a construction of A. E. Brouwer, we give examples to show that the bound is sharp for infinitely many values of n. This in turn gives an infinite family of examples which attain the bound given by H. Ehlich and by M. Wojtas for the determinant of a ± 1 matrix of dimension n ≡ 2 (mod 4). For n ≡ 3 (mod 4) we construct an infinite family of examples which attain slightly more than 1 3 of the bound given by Ehlich.