On Xia's construction of Hadamard difference sets
An Hadamard difference set is a difference set with parameters of the form \(v= 4u^2\), \(k= 2u^2\pm u\) and \(\lambda= u^2\pm u\). In a major paper, \textit{M. Y. Xia} [Some infinite classes of special Williamson matrices and difference sets, J. Comb. Theory, Ser. 61, No. 2, 230-242 (1992; Zbl 0772.05022)] proved the following existence result: Let \(p\) be a prime congruent to 3 modulo 4. Then there exists an abelian Hadamard difference set in the direct product of \(H\) with the elementary abelian group of order \(p^4\), where \(H\) is either of the two groups of order 4. However, Xia's proof used very complicated calculations involving cyclotomic classes of high order and was therefore hard to understand. The paper under review contains a character theoretic, much more transparent proof of this important result. It also points out an important connection to projective three-weight codes. This has by now led to similar constructions for some primes congruent to 1 modulo 4, i.e. \(p= 5, 13\) and 17; see the manuscripts by \textit{M. van Eupen} and \textit{V. D. Tonchev} [The existence of a reversible Hadamard difference set in \(Z_2\times Z_2\times Z_{5^4}\) and an infinite class of Williamson matrices] and \textit{R. M. Wilson} and \textit{Q. Xiang} [Constructions of Hadamard difference sets].
scientific article; zbMATH DE number 851863