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arXiv · 2607.20765

Multiplier obstructions for Legendre pairs of length 333

Arthur F. Ramos·David B. Hulak·Ruy J. G. B. de Queiroz
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Abstract

A Legendre pair of length 333 would yield a Hadamard matrix of order 668, the smallest order presently unresolved by the Hadamard conjecture. We study the structured case in which both sequences are fixed by a common subgroup $H\leq(\mathbb Z/333\mathbb Z)^\times$ acting by coordinate multiplication. We prove that such a pair can exist only when $|H|\leq 6$. After a mod-3 compression reduces the problem to an order-108 kernel, there are exactly 30 subgroups. We exclude 21 of them, including all 19 subgroups of order at least 9. The final order-9 subgroup is eliminated analytically: its orbit structure restricts the 9-compressed entries to $\{\pm1,\pm17,\pm19,\pm35,\pm37\}$; the Legendre equations force a $+17,-17$ pair in one compressed sequence, and a single-shift autocorrelation bound then contradicts the required compressed correlation. The remaining exclusions use full-image compression, a row-sum congruence, exact meet-in-the-middle enumeration, and proof-carrying pseudo-Boolean encodings. The solver-assisted cases are accompanied by independently checked DRAT proofs or direct arithmetic certificates. The result constrains fixed common-multiplier symmetry only; the unrestricted existence problems remain open.

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Arthur F. Ramos, David B. Hulak, Ruy J. G. B. de Queiroz. 2026-07-22. Multiplier obstructions for Legendre pairs of length 333. https://arxiv.org/abs/2607.20765

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