I‑AIMS Institute for Artificial Intelligence
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Hyperoctahedral Symmetry and Invariant Filtering for Turyn–Golay Quadruples

Ilias Kotsireas, Tanush Shaska

I-AIMS preprint sh-170 · 2026

We develop a group-theoretic framework for the study of quadruples of {−1, +1} sequences of length n whose non-periodic autocorrelations sum to zero at every nonzero shift, known in the literature as aperiodic complementary quadruples and as Turyn–Golay quadruples; their existence for every positive integer n implies the Hadamard conjecture. We prove that the group Eₙ of classical equivalence operations, of order 12288 and structure ((ℤ/2ℤ)² ≀ S₄) ⋊ ℤ/2ℤ, is the full stabilizer of the defining ideal inside the group Bₙ ≀ S₄ of signed permutations of the four sequences, where Bₙ = ℤ₂ⁿ ⋊ Sₙ is the hyperoctahedral group, and we recast existence as an orbit-intersection problem for an invariant algebraic variety Z. We prove unconditionally that the n − 1 defining forms are a regular sequence, so that they cut out a Gorenstein complete intersection subscheme of dimension 3n + 1. Invariant filtering via the second elementary symmetric function e₂, a Fourier-analytic reformulation as spectral flatness on the unit circle, and the decomposition of the module ℂ[{−1, 1}ⁿ] into Specht modules provide necessary existence conditions and computational search reductions. The anti-symmetry ansatz is interpreted as a restriction to a parabolic subgroup Bₘ × Bₖ ⊂ Bₙ, where the vanishing of head–tail cross-correlations is explained by a sign involution and, at the level of representations, by branching rules; Turyn’s doubling construction is shown to be an instance of this ansatz. The available sign twists are organized cohomologically through H¹(Bₙ, ℤ/2ℤ). We collect the known constructions for Turyn–Golay quadruples, as well as computational data on the spectrum of e₂ and on anti-symmetric solutions.

MSC classes: Primary 05B20, 94A55, 20C30; Secondary 20F55, 14G05, 13A50

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