Necessary conditions for menon difference sets

Abstract

Abstract. Menon difference sets have the parameters (4N 2, 2N 2 + N, N 2 5: N). In this paper, a necessary condition for Menon difference sets in groups of the form Z~p x Z2p • Gq, where Gq is an abelian q-group and p, q are distinct prime numbers, will be proved. We will also focus on the groups Z2p q X ~2pq and give constraints on the magnitudes ofp and q. Finally, we show that if the group Z6v • Z6p contains a Menon difference set, then p = 3 or 13 only.

New restrictions on possible orders of circulant Hadamard matrices

“…It is this method we will improve upon in the present paper. To make full use of our improvements, we will combine them with results of Turyn [13], McFarland [6], and Chan [3].…”

Cyclotomic integers and finite geometry

“…However, in this section we will consider arbitrary f and p. We will first deal with Hadamard difference sets. A lot of work has been devoted to finding necessary conditions for the existence of Hadamard difference sets; see [1,8,7,13,37,38,39,44,51]. However, all these results rely either on the self-conjugacy condition or on very restrictive assumptions on the parameter u.…”

The combinatorics of binary arrays

“…In recent years various authors have constructed perfect arrays for certain cases among those allowable cases listed above (see Arasu, Davis, Jedwab and Sehgal (1993), Davis (1991), Dillon (1990), Jedwab, Mitchell, Piper and Wild (1994)) or proved nonexistence in other cases (see Arasu and Jedwab (1992), Chan (1993), Chan and Siu (1991), Chan, Siu and Ma (1994), Jedwab (1991)). For an updated survey see Davis and Jedwab (1994).…”