`gr_mat_hadamard`

Synopsis#include <flint/gr_mat.h>

int gr_mat_hadamard(gr_mat_t res, gr_ctx_t ctx)source

Sets the square matrix res to a Hadamard matrix of its size if possible. Returns GR_DOMAIN if none exists (the size must be 0, 1, 2 or a multiple of 4), and GR_UNABLE if the implementation cannot construct one.

The matrix is constructed by fmpz_mat_hadamard and converted. It uses Sylvester's doubling and Paley's constructions [Pal1933], which succeed for all sizes $2^e$ and $2^e(q + 1)$ with $q$ an odd prime power. Sizes for which Hadamard matrices are known to exist but this construction fails start with 92, 116, 156 (OEIS A046116).

References#

[Pal1933] R. E. A. C. Paley, "On orthogonal matrices", Journal of Mathematics and Physics 12 (1933), 311–320. https://doi.org/10.1002/sapm1933121311

Cite this page

The FLINT team, “gr_mat_hadamard (gr_mat.h),” FLINT: Fast Library for Number Theory, documentation, version 3.7.0-dev, 2026. https://flintlib.org/doc/gr_mat/gr_mat_hadamard.html (generated 2026-11-02).

@misc{flint-doc-gr-mat-gr-mat-hadamard,
  author       = {{The FLINT team}},
  title        = {{gr\_mat\_hadamard (gr\_mat.h)}},
  howpublished = {FLINT: Fast Library for Number Theory, documentation, version 3.7.0-dev},
  year         = {2026},
  url          = {https://flintlib.org/doc/gr_mat/gr_mat_hadamard.html},
  note         = {Generated 2026-11-02}
}

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