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Returns true if and only if the matrix H is a Hadamard matrix.
Given a Hadamard matrix H, returns a normalized matrix equivalent to H.
This matrix is created by negating rows and columns to ensure that the
first row and column consist entirely of ones.
Given a Hadamard matrix H, returns a Hadamard-equivalent matrix H'
and transformation matrices X and Y such that H' = XHY.
The matrix H' is canonical in the sense that all matrices that are
Hadamard-equivalent to H (and no others) will produce the same
matrix H'.
Returns a sequence S of integers giving the 4-profile of the Hadamard
matrix H.
All Hadamard-equivalent matrices have the same 4-profile, but so may
some inequivalent ones.
Thus this test may determine inequivalence of two matrices more cheaply
than performing a full equivalence test, but cannot establish equivalence.
Al: MonStgElt Default: "nauty"
Returns true if and only if the Hadamard matrices H and J
are Hadamard equivalent.
The parameter Al may be set to either "Leon" or "nauty"
(the default).
If the "nauty" option is chosen and the matrices are equivalent
then transformation matrices X and Y are also returned, such that
J = XHY.
Returns an integer that encodes the entries in the given Hadamard matrix
in more compact form.
The intended use is to save time when testing for equality against the
same set of matrices many times.
Returns a Hadamard matrix of degree n whose encoded form is the
integer x.
This function is the inverse of HadamardMatrixToInteger().
We demonstrate the use of some of these functions on certain degree
16 Hadamard matrices. For convenience, we create them from the more
compact integer form.
> S := [
> 47758915483058652629300889924143904114798786457834842901517979108472281628672,
> 52517743516350345514775635367035460577743730272737765852723792818755052170805,
> 69809270372633075610047556428719374057869882804054059134346034969950931648512,
> 7209801227548712796135507135820555403251560090614832684136782016680445345792
> ];
> T := [ HadamardMatrixFromInteger(x, 16) : x in S ];
> &and [ IsHadamard(m) : m in T ];
true
Now we can test them for equivalence; we start by checking the 4-profiles.
> [ HadamardInvariant(m) : m in T ];
[
[ 1680, 0, 140 ],
[ 1680, 0, 140 ],
[ 1344, 448, 28 ],
[ 1344, 448, 28 ]
]
We see that the only possible equivalencies are between the first two
and the last two, since equivalent matrices must have the same 4-profile.
> equiv,X,Y := IsHadamardEquivalent(T[1], T[2]);
> equiv;
true
> T[2] eq X*T[1]*Y;
true
> equiv,X,Y := IsHadamardEquivalent(T[3], T[4]);
> equiv;
false
So we have three inequivalent matrices.
An alternative way to determine the inequivalent matrices would
have been to use the canonical forms.
> #{ HadamardCanonicalForm(m) : m in T };
3
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