Recent progress in the construction of Hadamard matrices

@misc{egan2026recent,
  title = {{Recent progress in the construction of Hadamard matrices}},
  author = {Ronan Egan and Ilias Kotsireas and Stephen London and Pádraig Ó Catháin},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.27175},
  eprint = {2610.27175},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.27175},
  note = {CC BY 4.0},
}

TY  - UNPB
TI  - Recent progress in the construction of Hadamard matrices
AU  - Ronan Egan
AU  - Ilias Kotsireas
AU  - Stephen London
AU  - Pádraig Ó Catháin
PY  - 2026
DA  - 2026/10/19
PB  - arXiv
DO  - 10.48550/arxiv.2610.27175
UR  - https://arxiv.org/abs/2610.27175
AB  - A Hadamard matrix is a square matrix with entries ± 1 whose rows are mutually orthogonal. Hadamard matrices are conjectured to exist at every order divisible by $4$; for approximately the last twenty years, the smallest open case of this conjecture was order 668. In August 2026, Alpöge, Voinov and Reynolds-Haertle announced Hadamard matrices at the twelve orders below $2000$ at which none was known. In this paper we verify and analyse the claimed matrices. We survey results on the existence of Hadamard matrices and list the $95$ orders up to 10 000 at which no Hadamard matrix is currently known. In pursuing this work we constructed Hadamard matrices at sixteen orders that were open in the published literature. We conclude with a list of open problems.
ER  -