# Generalized constructions of Menon-Hadamard difference sets

## TL;DR

This paper broadens the methods for constructing Menon-Hadamard difference sets by generalizing previous frameworks, introducing flexible constructions using cyclotomic classes and solving an open problem from 1997.

## Contribution

It generalizes Chen's construction with semi-primitive cyclotomic classes and provides a new, more flexible construction of spreads and projective sets of type Q in PG(3,q).

## Key findings

- Demonstrates greater flexibility in constructing projective sets of type Q.
- Provides a new construction of spreads and projective sets for all odd prime powers q.
- Solves an open problem from Wilson-Xiang 1997.

## Abstract

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 ${\mathrm{PG}}(3,q)$. 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 ${\mathrm{PG}}(3,q)$ 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 $4q^4$ 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 ${\mathrm{PG}}(3,q)$ 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.