Constructions of skew Hadamard difference families. (February 2019)

• Record Type: Journal Article
• Authors: Momihara, Koji; Xiang, Qing
• Is Part Of: European journal of combinatorics. Volume 76(2019)
• Page Start: 73
• Page End: 81
• Publication Date: 2019-02
• Subjects: Combinatorial analysis -- Periodicals; Analyse combinatoire -- Périodiques; Combinatorial analysis; Periodicals; Electronic journals; 511.6
• DOI: 10.1016/j.ejc.2018.09.007
• Languages: English
• ISSNs: 0195-6698

Abstract

Abstract: In this paper, we generalize classical constructions of skew Hadamard difference families with two or four blocks in the additive groups of finite fields given by Szekeres (1969, 1971), Whiteman (1971) and Wallis–Whiteman (1972). In particular, we show that there exists a skew Hadamard difference family with 2 u − 1 blocks in the additive group of the finite field of order q e for any prime power q ≡ 2 u + 1 ( mod 2 u + 1 ) with u ⩾ 2 and any positive integer e . In the aforementioned papers of Szekeres, Whiteman, and Wallis–Whiteman, the constructions of skew Hadamard difference families with 2 u − 1 ( u = 2 or 3) blocks in ( F q e, + ) work only for restricted e ;namely e ≡ 1, 2, or 3 ( mod 4 ) when u = 2, and e ≡ 1 ( mod 2 ) when u = 3, respectively. Our more general construction, in particular, removes the restrictions on e . As a consequence, we obtain new infinite series of skew Hadamard difference families with two or four blocks, and hence skew Hadamard matrices.