Corrigendum to ``Barker sequences and difference sets''
Wayne Broughton has pointed out to us that Lemma 2 [Enseign. Math., II. Sér. 38, No. 3/4, 345-382 (1992; Zbl 0777.05024)] in our proof of Turyn's theorem (which says that, if \(D\subset \mathbb{Z}/v\mathbb{Z}\) is a cyclic difference set with parameters \((v,k,\lambda)= (4N^ 2, 2N^ 2- N, N^ 2-N)\), then \(N\) is odd) is incorrect. Indeed, for \(r=3\), \(s=2\), the element \(\alpha= 1+ x^ 4_ 3\in \mathbb{Z}\Gamma_ 3\) yields a counterexample. The correct version of Lemma 2 should read as follows: Lemma 2. Let \(\alpha\in \mathbb{Z}\Gamma_ r\) \((r\geq 1)\) be such that, for some \(s\leq r\), \(\rho v_ j(\alpha)\in 2^{j+1}\mathbb{Z}[\eta_{r-j}]\) for all \(j= 0,\dots,s-1\). Then, \(v_ s(\alpha)\in 2^ s\mathbb{Z}\Gamma_{r- s}\).
scientific article; zbMATH DE number 637639