2019/05/21 by Koji Momihara, Momihara, Koji, Qing Xiang +1
Computer Science · Engineering · Mathematics · #Coding theory and cryptography #Combinatorics (math.CO) #FOS: Mathematics #Finite Group Theory Research #graph theory and CDMA systems
paper · pdf · doi:10.48550/arxiv.1905.08470
openalex publication_date 2019/05/21 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We revisit the problem of constructing Menon-Hadamard difference sets. In 1997, Wilson and Xiang gave a general framework for constructing Menon-Hadamard difference sets by using a combination of a spread and four projective sets of type Q in PG(3,q). They also found examples of suitable spreads and projective sets of type Q for q=5,13,17. Subsequently, Chen (1997) succeeded in finding a spread and four projective sets of type Q in PG(3,q) satisfying the conditions in the Wilson-Xiang construction for all odd prime powers q. Thus, he showed that there exists a Menon-Hadamard difference set of order 4q4 for all odd prime powers q. However, the projective sets of type Q found by Chen have automorphisms different from those of the examples constructed by Wilson and Xiang. In this paper, we first generalize Chen's construction of projective sets of type Q by using `semi-primitive' cyclotomic classes. This demonstrates that the construction of projective sets of type Q satisfying the conditions in the Wilson-Xiang construction is much more flexible than originally thought. Secondly, we give a new construction of spreads and projective sets of type Q in PG(3,q) for all odd prime powers q, which generalizes the examples found by Wilson and Xiang. This solves a problem left open in Section 5 of the Wilson-Xiang paper from 1997.