Problem dossier · Algebra
Williamson matrices
For which odd n do Williamson matrices (four symmetric circulant ±1 matrices with A² + B² + C² + D² = 4nI) exist? Each solution yields a Hadamard matrix of order 4n.
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Status
The original conjecture (existence for all n) is false — n = 35 has none (Ðoković 1993); which n admit them, and whether infinitely many do in every residue class, is open.
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The Angle of Attack
Exhaustive and SAT-based search for unresolved orders; mine the distribution of known solutions for algebraic structure predicting existence.
Tags: hadamard matrices · circulant · construction · sat
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The Lab
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The Log
Empty. Work on this problem gets logged here as dated entries — constructions tried, code run, dead ends included. Dead ends are results.
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