Cyclotomic Fields in Finite Geometry 2025, 35 - Self-Conjugacy Exponent Bounds

Let G be an abelian group of order v, n a positive integer with (v, n) > 1, p a prime divisor of (v, n) which is self-conjugate module expG with p^2a dividing n for some a ≥ 1. If D ∈ ℤ[G] with nonnegative coeff, bounded by constant C so that DD^(-1) = n + AW, for A ∈ ℤ[G] and W ⊂ G of order div by p. For G^p Sylow p-subgroup of G, then |G^p|/expG_p ≥ p^a/C. With a (v, k, λ, n)-difference set in G, a normal subgroup N of G with G/N abelian and p self-conjugate mod expG/N, let P be the Sylow p-subgroup of G/N, then |P|/expP ≥ p^a/|N|.

Let D = Σ_g∈Ga_gg ∈ ℤ[G]: a_g ≥ 0 ∀ g, a∈P, A = ⟨b⟩×W a subgroup of P with A ∩ ⟨a⟩ = {1}, o(a) = p^t ≥ exp A and o(b) ≥ p, with δ some positive integer and U ∈ S(a, A), g∈G either

• D(Ug) - D(Uga^pt-1)≥δ and D(Uga^ipt-1)<δ data-preserve-html-node="true"/p for i = 1, ..., p - 1
• D(Ug) < δ/p and at least one coset Ug satisfying the first condition, write B = ⟨b^p⟩×W, the ∀ U' ∈ S(a, B), the coset U'g satisfies one of the conditions and there is at least one satisfying the first.

A McFarland difference set has explicit parameters for q = p^f ≠ 2, p prime,

• v = q^d+1[1+(q^d+1-1)/(q-1)]
• k = q^d(q^d+1-1)/(q-1)
• λ = q^d(q^d-1)/(q-1)
• n = q^2d

Its defined by the triple (p, f, d). For q = 2, the set is Hadamard. A McFarland difference set D in a abelian group G of order q^d+1[1 + (q^d+1 - 1)/(q - 1)] where q = p^f and p is a prime self-conjugate modula expG. With Sylow p-subgroup of G, if p is odd, P is elementary abelian. If p = 2, f ≥ 2, then expP ≤ 4. If p is odd, then a McFarland difference set in G exists iff expP = p. If p = f = 2, then McFarland difference set in G exists iff expP ≤ 4. If G is abelian with order p^2a+b, N its subgroup of order p^b, and there is a (p^2a, p^b, p^2a, p^2a-b) difference set in G relative to N, then expN ≤ p^a. If p is an odd prime, then expG ≤ p└^a/2┘⁺¹.