Multiplier obstructions for Legendre pairs of length 333

@misc{ramos2026multiplier,
  title = {{Multiplier obstructions for Legendre pairs of length 333}},
  author = {Arthur F. Ramos and David B. Hulak and Ruy J. G. B. de Queiroz},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2607.20765},
  eprint = {2607.20765},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2607.20765},
}

TY  - UNPB
TI  - Multiplier obstructions for Legendre pairs of length 333
AU  - Arthur F. Ramos
AU  - David B. Hulak
AU  - Ruy J. G. B. de Queiroz
PY  - 2026
DA  - 2026/10/28
PB  - arXiv
DO  - 10.48550/arxiv.2607.20765
UR  - https://arxiv.org/abs/2607.20765
AB  - 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≤(ℤ/333ℤ)^× acting by coordinate multiplication. We prove that such a pair can exist only when |H|≤ 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 {±1,±17,±19,±35,±37}; 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.
ER  -