Difference families from lines and half lines
The author generalizes constructions of \textit{M.-Y. Xia} and \textit{G. Liu} [``An infinite class of supplementary difference sets and Williamson matrices'', J. Comb. Theory, Ser. A 58, No. 2, 310-317 (1991; Zbl 0758.05032); ``A new family of supplementary difference sets and Hadamard matrices'', J. Stat. Plann. Inference 51, No. 3, 283-291 (1996; Zbl 0851.05027)] to construct some new \((v,k,\lambda)\)-difference families, with parameters (i) \(v= q^2\), \(k= (q^2- q)/2\) and \(\lambda= (eq^2- 2eq)/2\), where \(q\equiv e-1\pmod{2e}\) is a prime power, \(e\equiv 2\pmod 4\); and (ii) \(v= q^2\), \(k= (q^2- q)/2\) and \(\lambda= (eq^2- 2eq)/4\), where \(q\equiv e-1\pmod{2e}\) is a prime power, \(e\equiv 0\pmod 4\). Here the smallest cases \(e= 2\) resp. \(e=4\) are due to Xia and Liu; these cases can be used to construct Williamson and Hadamard matrices. Xiang's construction makes use of the \(2e\)th cyclotomic classes, but uses a geometric interpretation in terms of lines and half-lines; this results in considerably simpler proofs than those given by Xia and Liu.
scientific article; zbMATH DE number 1199960