Cyclotomic Fields in Finite Geometry 2025, 29 - Hadamard Matrix Example

Hadamard matrices of order v are v × v matrices with entries ±1 and two orthogonal rows. The problem this book addresses is exemplified by seeking a Hadamard matrix invariant to cyclic shifting, thus qualifying it as "circulant". The Circulant Hadamard matrix conjecture is that there are no circulant Hadamard matrices of order ≠ 1, 4. Assume a circulant Hadamard matrix H = (h_i,j)_i,j=0^v-1 of order v. It fulfills h_i+1,j+1 = h_i,j ↠ h_i+k,j+k=h_i,j. Define X := Σ_i=0^v-1 h_iξⁱ which has modulus equal to v. From here, this opens a combinatoric question of whether this is true for arbitrary v. We can set X ∈ ℤ[ξ] and its principal ideal I := (X) with |I|² = (v).

For positive integers m, n, k, λ with mn ≥ k ≥ 2, a divisible (m, n, k λ) design D = (P, B) is a set of points P with a set of subsets B of P blocks so that each block has exactly k points and the point set P can be partitioned into m classes of size n so that any two are contained in exactly λ common blocks if they're in different point classes and contained in no common block if they are in the same point class. (p, b) with p ∈ b ∈ B is a flag. If n = 1, then write v instead of m and (v, k, λ) design. Any 2 points are contained in exactly λ common blocks. If such a design exists, then λ(v - 1) = 0 (mod k-1) and λv(v-1) = 0 (mod k(k - 1)). For fixed k, λ the conditions for the existence of a (v, k λ) design are sufficient for all v greater than some constant depending on k, λ. If the number of points is equal to the number of blocks, the design is called a "symmetric" (better: "square"). Point/block regular automorphism groups of a divisible design D if for any two points in P/blocks in B there is exactly one automorphism in its automorphism group with τ(p) = q(τ(B) = C). An automorphism group of a symmetric (v, k, λ) design is point regular iff it's block regular.

For a finite group G of order nm with subgroup N of order n, a subset R of G is a (m, n, k, λ) difference set in G relative to N if ∀ g∈G\N ∃! λ representations g=r₁r₂^-1, r₁, r₂ ∈ R, and no nonidentity element of N has such a representation. The subgroup N is the "forbidden subgroup". In the case n=1, then write (v, k, λ) difference set in G. The non-negative n = k - λ is the order of the difference set. If it's either 0 or 1, the difference set is "trivial". For an existence of such an R, then D = (G, {Rg: g ∈ G}) is a symmetric divisible (m, n, k, λ) design, and G is a point-regular automorphism group of D by right translation. The point classes of D are the cosets of N. A (v, k λ) difference set is equiv. to a symmetric (v, k, λ) design with G as a point-regular automorphism group.

A prime power q, with integer d ≥ 2, then D :={x𝔽*_q: x ∈ 𝔽*_q^d+1, Tr(x) = 0} is a difference set in G := 𝔽*_q^d+1/𝔽*_q with the classic point-hyperplane design. Note PG(d, q) as the set of all subspaces of V is called the Desarguesian projective geometry of dim d over 𝔽_q. The point-regular automorphism group over it is the Singer cycle of PG(d, q), invoking the Singer difference set of PG(d, q).

Returning to the Hadamard matrices, if a Hadamard matrix of order v exists, then v = 1, 2, or v = 0 (mod 4). It's assumed that there is a Hadamard matrix of order v for all v = 0 (mod 4). As a generalization, Hadamard matrices may be abstracted to weighing matrices W(m, n) which are m × m matrices with entries ±1, 0 with HH^T = nI for some nonnegative integer n. n is referred to as the "weight" of H. The attention to weighing matrices which are invariant under a group operation, then G is a group of order n. If a G-invariant weighing matrix H = W(m, n) exists, then n is a square of an integer s, and the number of entries 1 in each row of H is s(s+1)/2.