Nonexistence of Hadamard difference sets in groups of order 4p² with p≡ 3 pmod 4, p≥ 7

@misc{schmidt2026nonexistence,
  title = {{Nonexistence of Hadamard difference sets in groups of order $4p^2$ with $p\equiv 3 \pmod 4$, $p\ge 7$}},
  author = {Bernhard Schmidt},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.37417},
  eprint = {2610.37417},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.37417},
}

TY  - UNPB
TI  - Nonexistence of Hadamard difference sets in groups of order 4p^2 with p≡ 3 pmod 4, p≥ 7
AU  - Bernhard Schmidt
PY  - 2026
DA  - 2026/10/27
PB  - arXiv
DO  - 10.48550/arxiv.2610.37417
UR  - https://arxiv.org/abs/2610.37417
AB  - We show that the Frobenius group ⟨ x,y,z| x^p=y^p=z^4=1, xy=yx, zxz^−1=y, zyz^−1=x^−1⟩ of order 4p^2 contains no (4p^2,2p^2−p,p^2−p) difference set if p≡ 3pmod 4 is a prime with p ≥ 7. Together with results of McFarland, Iiams and Liebler this shows that no group of order 4p^2 with p ≡ 3 pmod 4 and p ≥ 7 contains a Hadamard difference set.
ER  -