The maximum determinant of \(\pm 1\) matrices (Q1355220)

The authors give a new proof for the bound on the value of the determinant of a \(\pm1\) matrix of dimension \(n\equiv 1\pmod 4\) first given by \textit{G. Barba} [Giorn. Mat. Battaglini, III. Ser. 71, 70-86 (1933; Zbl 0007.39102)]. Adapting a construction of \textit{A. E. Brouwer} [Math. Cent., Amst., Afd. Zuivere wiskd. ZW 202/83, 5 p. (1983; Zbl 0534.05011)], examples are given, showing 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 \textit{H. Ehlich} [Math. Z. 83, 123-132 (1964; Zbl 0115.24704)] and by \textit{M. Wojtas} [Colloq. Math. 12, 73-83 (1964; Zbl 0126.02604)] for the determinant of a \(\pm1\) matrix of dimension \(n\equiv 2\pmod 4\). For \(n\equiv 3\pmod 4\) the authors construct an infinite family of examples which attain slightly more than \(1/3\) of the bound given by Ehlich [loc. cit.].