Hadamard matrices

A Hadamard matrix is an \(n\times n\) matrix \(H\) whose entries are either \(+1\) or \(-1\) and whose rows are mutually orthogonal. For example, the matrix \(H_2\) defined by

\[\begin{split}\left(\begin{array}{rr} 1 & 1 \\ 1 & -1 \end{array}\right)\end{split}\]

is a Hadamard matrix. An \(n\times n\) matrix \(H\) whose entries are either \(+1\) or \(-1\) is a Hadamard matrix if and only if:

1. \(|det(H)|=n^{n/2}\) or

2. \(H*H^t = n\cdot I_n\), where \(I_n\) is the identity matrix.

In general, the tensor product of an \(m\times m\) Hadamard matrix and an \(n\times n\) Hadamard matrix is an \((mn)\times (mn)\) matrix. In particular, if there is an \(n\times n\) Hadamard matrix then there is a \((2n)\times (2n)\) Hadamard matrix (since one may tensor with \(H_2\)). This particular case is sometimes called the Sylvester construction.

The Hadamard conjecture (possibly due to Paley) states that a Hadamard matrix of order \(n\) exists if and only if \(n= 1, 2\) or \(n\) is a multiple of \(4\).

The module below implements constructions of Hadamard and skew Hadamard matrices for all known orders \(\le 1200\), plus some more greater than \(1200\). It also allows you to pull a Hadamard matrix from the database at [SloaHada].

The following code will test that a construction for all known orders \(\le 4k\) is implemented. The assertion above can be verified by setting k=300 (note that it will take a long time to run):

sage: from sage.combinat.matrices.hadamard_matrix import (hadamard_matrix,
....:                               skew_hadamard_matrix, is_hadamard_matrix,
....:                               is_skew_hadamard_matrix)
sage: k = 20
sage: unknown_hadamard = [668, 716, 892, 1132]
sage: unknown_skew_hadamard = [356, 404, 428, 476, 596, 612, 668, 708, 712, 716,
....:                         764, 772, 804, 808, 820, 836, 856, 892, 900, 916,
....:                         932, 940, 952, 980, 996, 1004, 1012, 1028, 1036,
....:                         1044, 1060, 1076, 1100, 1108, 1132, 1140, 1148,
....:                         1156, 1180, 1192, 1196]
sage: for n in range(1, k+1):
....:   if 4*n not in unknown_hadamard:
....:       H = hadamard_matrix(4*n, check=False)
....:       assert is_hadamard_matrix(H)
....:   if 4*n not in unknown_skew_hadamard:
....:       H = skew_hadamard_matrix(4*n, check=False)
....:       assert is_skew_hadamard_matrix(H)

AUTHORS:

• David Joyner (2009-05-17): initial version

• Matteo Cati (2023-03-18): implemented more constructions for Hadamard and skew Hadamard matrices, to cover all known orders up to 1200.

REFERENCES:

• [SloaHada]

• [HadaWiki]

• [Hora]

• [CP2023]

sage.combinat.matrices.hadamard_matrix.GS_skew_hadamard_smallcases(n, existence=False, check=True)

Data for Williamson-Goethals-Seidel construction of skew Hadamard matrices

Here we keep the data for this construction. Namely, it needs 4 circulant matrices with extra properties, as described in sage.combinat.matrices.hadamard_matrix.williamson_goethals_seidel_skew_hadamard_matrix() Matrices are taken from:

• \(n=36, 52\): [GS70s]

• \(n=92\): [Wall71]

• \(n=188\): [Djo2008a]

• \(n=236\): [FKS2004]

• \(n=276\): [Djo2023a]

Additional data is obtained from skew supplementary difference sets contained in sage.combinat.designs.difference_family.skew_supplementary_difference_set(), using the construction described in [Djo1992a].

INPUT:

• n – integer; the order of the matrix

• existence – boolean (default: True); if True, only check that we can do the construction

• check – boolean (default: False): if True, check the result

sage.combinat.matrices.hadamard_matrix.RSHCD_324(e)

Return a size 324x324 Regular Symmetric Hadamard Matrix with Constant Diagonal.

We build the matrix \(M\) for the case \(n=324\), \(\epsilon=1\) directly from JankoKharaghaniTonchevGraph and for the case \(\epsilon=-1\) from the “twist” \(M'\) of \(M\), using Lemma 11 in [HX2010]. Namely, it turns out that the matrix

\[\begin{split}M'=\begin{pmatrix} M_{12} & M_{11}\\ M_{11}^\top & M_{21} \end{pmatrix}, \quad\text{where}\quad M=\begin{pmatrix} M_{11} & M_{12}\\ M_{21} & M_{22} \end{pmatrix},\end{split}\]

and the \(M_{ij}\) are 162x162-blocks, also RSHCD, its diagonal blocks having zero row sums, as needed by [loc.cit.]. Interestingly, the corresponding \((324,152,70,72)\)-strongly regular graph has a vertex-transitive automorphism group of order 2592, twice the order of the (intransitive) automorphism group of the graph corresponding to \(M\). Cf. [CP2016].

INPUT:

• e – \(-1\) or \(+1\); the value of \(\epsilon\)

REFERENCE:

• [CP2016]

sage.combinat.matrices.hadamard_matrix.amicable_hadamard_matrices(n, existence=False, check=True)

Construct amicable Hadamard matrices of order n using the available methods.

INPUT:

• n – positive integer; the order of the amicable Hadamard matrices

• existence – boolean (default: False); if True, only return whether amicable Hadamard matrices of order \(n\) can be constructed

• check – boolean (default: True); if True, check that the matrices are amicable Hadamard matrices before returning them

OUTPUT:

If existence is true, the function returns a boolean representing whether amicable Hadamard matrices of order \(n\) can be constructed. If existence is false, returns two amicable Hadamard matrices, or raises an error if the matrices cannot be constructed.

EXAMPLES:

sage: from sage.combinat.matrices.hadamard_matrix import amicable_hadamard_matrices
sage: amicable_hadamard_matrices(2)
(
[ 1  1]  [ 1  1]
[-1  1], [ 1 -1]
)

If existence is true, the function returns a boolean:

sage: amicable_hadamard_matrices(16, existence=True)
False

sage.combinat.matrices.hadamard_matrix.amicable_hadamard_matrices_wallis(n, check=True)

Construct amicable Hadamard matrices of order \(n = q + 1\) where \(q\) is a prime power.

If \(q\) is a prime power \(\equiv 3 \mod 4\), then amicable Hadamard matrices of order \(q+1\) can be constructed as described in [Wal1970b].

INPUT:

• n – integer; the order of the matrices to be constructed

• check – boolean (default: True); if True, check that the resulting matrices are amicable Hadamard before returning them

OUTPUT:

The function returns two amicable Hadamard matrices, or raises an error if such matrices cannot be created using this construction.

EXAMPLES:

sage: from sage.combinat.matrices.hadamard_matrix import amicable_hadamard_matrices_wallis
sage: M, N = amicable_hadamard_matrices_wallis(28)

sage.combinat.matrices.hadamard_matrix.are_amicable_hadamard_matrices(M, N, verbose=False)

Check if M and N are amicable Hadamard matrices.

Two matrices \(M\) and \(N\) of order \(n\) are called amicable if they satisfy the following conditions (see [Seb2017]):

• \(M\) is a skew Hadamard matrix

• \(N\) is a symmetric Hadamard matrix

• \(MN^T = NM^T\)

INPUT:

• M – a square matrix

• N – a square matrix

• verbose – boolean (default False); whether to be verbose when the matrices are not amicable Hadamard matrices

EXAMPLES:

sage: from sage.combinat.matrices.hadamard_matrix import are_amicable_hadamard_matrices
sage: M = matrix([[1, 1], [-1, 1]])
sage: N = matrix([[1, 1], [1, -1]])
sage: are_amicable_hadamard_matrices(M, N)
True

If verbose is true, the function will be verbose when returning False:

sage: N = matrix([[1, 1], [1, 1]])
sage: are_amicable_hadamard_matrices(M, N, verbose=True)
The second matrix is not Hadamard
False

sage.combinat.matrices.hadamard_matrix.construction_four_symbol_delta_code_I(X, Y, Z, W)

Construct 4-symbol \(\delta\) code of length \(2n+1\).

The 4-symbol \(\delta\) code is constructed from sequences \(X, Y, Z, W\) of length \(n+1\), \(n+1\), \(n\), \(n\) satisfying for all \(s > 0\):

\[N_X(s) + N_Y(s) + N_Z(s) + N_W(s) = 0\]

where \(N_A(s)\) is the nonperiodic correlation function:

\[N_A(s) = \sum_{i=1}^{n-s}a_ia_{i+s}\]

The construction (detailed in [Tur1974]) is as follows:

\[\begin{split}\begin{aligned} T_1 &= X;Z \\ T_2 &= X;-Z \\ T_3 &= Y;W \\ T_4 &= Y;-W \end{aligned}\end{split}\]

INPUT:

• X – list; the first sequence (length \(n+1\))

• Y – list; the second sequence (length \(n+1\))

• Z – list; the third sequence (length \(n\))

• W – list; the fourth sequence (length \(n\))

OUTPUT:

A tuple containing the 4-symbol \(\delta\) code of length \(2n+1\).

EXAMPLES:

sage: from sage.combinat.matrices.hadamard_matrix import construction_four_symbol_delta_code_I
sage: construction_four_symbol_delta_code_I([1, 1], [1, -1], [1], [1])
([1, 1, 1], [1, 1, -1], [1, -1, 1], [1, -1, -1])

sage.combinat.matrices.hadamard_matrix.construction_four_symbol_delta_code_II(X, Y, Z, W)

Construct 4-symbol \(\delta\) code of length \(4n+3\).

The 4-symbol \(\delta\) code is constructed from sequences \(X, Y, Z, W\) of length \(n+1\), \(n+1\), \(n\), \(n\) satisfying for all \(s > 0\):

\[N_X(s) + N_Y(s) + N_Z(s) + N_W(s) = 0\]

where \(N_A(s)\) is the nonperiodic correlation function:

\[N_A(s) = \sum_{i=1}^{n-s}a_ia_{i+s}\]

The construction (detailed in [Tur1974]) is as follows (writing \(A/B\) to mean \(A\) alternated with \(B\)):

\[\begin{split}\begin{aligned} T_1 &= X/Z;Y/W;1 \\ T_2 &= X/Z;Y/-W;-1 \\ T_3 &= X/Z;-Y/-W;1 \\ T_4 &= X/Z;-Y/W;-1 \end{aligned}\end{split}\]

INPUT:

• X – list; the first sequence (length \(n+1\))

• Y – list; the second sequence (length \(n+1\))

• Z – list; the third sequence (length \(n\))

• W – list; the fourth sequence (length \(n\))

OUTPUT:

A tuple containing the four 4-symbol \(\delta\) code of length \(4n+3\).

EXAMPLES:

sage: from sage.combinat.matrices.hadamard_matrix import construction_four_symbol_delta_code_II
sage: construction_four_symbol_delta_code_II([1, 1], [1, -1], [1], [1])
([1, 1, 1, 1, 1, -1, 1],
 [1, 1, 1, 1, -1, -1, -1],
 [1, 1, 1, -1, -1, 1, 1],
 [1, 1, 1, -1, 1, 1, -1])

sage.combinat.matrices.hadamard_matrix.four_symbol_delta_code_smallcases(n, existence=False)

Return the 4-symobl \(\delta\) code of length n if available.

The 4-symbol \(\delta\) codes are constructed using construction_four_symbol_delta_code_I() or construction_four_symbol_delta_code_II(). The base sequences used are taken from [Tur1974].

INPUT:

• n – integer; the length of the desired 4-symbol \(\delta\) code

• existence – boolean (default: False); if True, only check if the sequences are available

EXAMPLES:

sage: from sage.combinat.matrices.hadamard_matrix import four_symbol_delta_code_smallcases
sage: four_symbol_delta_code_smallcases(3)
([1, -1, 1], [1, -1, -1], [1, 1, 1], [1, 1, -1])
sage: four_symbol_delta_code_smallcases(3, existence=True)
True

sage.combinat.matrices.hadamard_matrix.hadamard_matrix(n, existence=False, check=True)

This function is available as hadamard_matrix(…) and matrix.hadamard(…).

Tries to construct a Hadamard matrix using the available methods.

Currently all orders \(\le 1200\) for which a construction is known are implemented. For \(n > 1200\), only some orders are available.

INPUT:

• n – integer; dimension of the matrix

• existence – boolean (default: False); whether to build the matrix or merely query if a construction is available in Sage. When set to True, the function returns:

  • True – meaning that Sage knows how to build the matrix

  • Unknown – meaning that Sage does not know how to build the matrix, although the matrix may exist (see sage.misc.unknown).

  • False – meaning that the matrix does not exist.

• check – boolean (default: True); whether to check that output is correct before returning it. As this is expected to be useless (but we are cautious guys), you may want to disable it whenever you want speed.

EXAMPLES:

sage: hadamard_matrix(12).det()
2985984
sage: 12^6
2985984
sage: hadamard_matrix(1)
[1]
sage: hadamard_matrix(2)
[ 1  1]
[ 1 -1]
sage: hadamard_matrix(8) # random
[ 1  1  1  1  1  1  1  1]
[ 1 -1  1 -1  1 -1  1 -1]
[ 1  1 -1 -1  1  1 -1 -1]
[ 1 -1 -1  1  1 -1 -1  1]
[ 1  1  1  1 -1 -1 -1 -1]
[ 1 -1  1 -1 -1  1 -1  1]
[ 1  1 -1 -1 -1 -1  1  1]
[ 1 -1 -1  1 -1  1  1 -1]
sage: hadamard_matrix(8).det() == 8^4
True

We note that hadamard_matrix() returns a normalised Hadamard matrix (the entries in the first row and column are all +1)

sage: hadamard_matrix(12) # random
[ 1  1| 1  1| 1  1| 1  1| 1  1| 1  1]
[ 1 -1|-1  1|-1  1|-1  1|-1  1|-1  1]
[-----+-----+-----+-----+-----+-----]
[ 1 -1| 1 -1| 1  1|-1 -1|-1 -1| 1  1]
[ 1  1|-1 -1| 1 -1|-1  1|-1  1| 1 -1]
[-----+-----+-----+-----+-----+-----]
[ 1 -1| 1  1| 1 -1| 1  1|-1 -1|-1 -1]
[ 1  1| 1 -1|-1 -1| 1 -1|-1  1|-1  1]
[-----+-----+-----+-----+-----+-----]
[ 1 -1|-1 -1| 1  1| 1 -1| 1  1|-1 -1]
[ 1  1|-1  1| 1 -1|-1 -1| 1 -1|-1  1]
[-----+-----+-----+-----+-----+-----]
[ 1 -1|-1 -1|-1 -1| 1  1| 1 -1| 1  1]
[ 1  1|-1  1|-1  1| 1 -1|-1 -1| 1 -1]
[-----+-----+-----+-----+-----+-----]
[ 1 -1| 1  1|-1 -1|-1 -1| 1  1| 1 -1]
[ 1  1| 1 -1|-1  1|-1  1| 1 -1|-1 -1]

sage.combinat.matrices.hadamard_matrix.hadamard_matrix_156()

Construct a Hadamard matrix of order 156.

The matrix is created using the construction detailed in [BH1965]. This uses four circulant matrices of size \(13\times 13\), which are composed into a \(156\times 156\) block matrix.

sage.combinat.matrices.hadamard_matrix.hadamard_matrix_cooper_wallis_construction(x1, x2, x3, x4, A, B, C, D, check=True)

Create a Hadamard matrix using the contruction detailed in [CW1972].

Given four circulant matrices \(X_1\), X_2, X_3, X_4` of order \(n\) with entries (0, 1, -1) such that the entrywise product of two distinct matrices is always equal to \(0\) and that \(\sum_{i=1}^{4}X_iX_i^\top = nI_n\) holds, and four matrices \(A, B, C, D\) of order \(m\) with elements (1, -1) such that \(MN^\top = NM^\top\) for all distinct \(M\), \(N\) and \(AA^\top + BB^\top + CC^\top + DD^\top = 4mI_n\) holds, we construct a Hadamard matrix of order \(4nm\).

INPUT:

• x1 – list or vector; the first row of the circulant matrix \(X_1\)

• x2 – list or vector; the first row of the circulant matrix \(X_2\)

• x3 – list or vector; the first row of the circulant matrix \(X_3\)

• x4 – list or vector; the first row of the circulant matrix \(X_4\)

• A – the matrix described above

• B – the matrix described above

• C – the matrix described above

• D – the matrix described above

• check – boolean (default: True); if True, check that the resulting matrix is Hadamard before returing it.

EXAMPLES:

sage: from sage.combinat.matrices.hadamard_matrix import hadamard_matrix_cooper_wallis_construction
sage: from sage.combinat.t_sequences import T_sequences_smallcases
sage: seqs = T_sequences_smallcases(19)
sage: hadamard_matrix_cooper_wallis_construction(seqs[0], seqs[1], seqs[2], seqs[3], matrix([1]), matrix([1]), matrix([1]), matrix([1]))
76 x 76 dense matrix over Integer Ring...

sage.combinat.matrices.hadamard_matrix.hadamard_matrix_cooper_wallis_smallcases(n, check=True, existence=False)

Construct Hadamard matrices using the Cooper-Wallis construction for some small values of \(n\).

This function calls the function hadamard_matrix_cooper_wallis_construction() with the appropriate arguments. It constructs the matrices \(X_1\), \(X_2\), \(X_3\), \(X_4\) using either T-matrices or the T-sequences from sage.combinat.t_sequences.T_sequences_smallcases(). The matrices \(A\), \(B\), \(C\), \(D\) are taken from williamson_type_quadruples_smallcases().

Data for T-matrices of order 67 is taken from [Saw1985].

INPUT:

• n – integer; the order of the matrix to be constructed

• check – boolean (default: True); if True, check that the matrix is a Hadamard matrix before returning

• existence – boolean (default: False); if True, only check if the matrix exists.

OUTPUT:

If existence=False, returns the Hadamard matrix of order \(n\). It raises an error if no data is available to construct the matrix of the given order. If existence=True, returns a boolean representing whether the matrix can be constructed or not.

See also

hadamard_matrix_cooper_wallis_construction()

EXAMPLES:

By default The function returns the Hadamard matrix

sage: from sage.combinat.matrices.hadamard_matrix import hadamard_matrix_cooper_wallis_smallcases
sage: hadamard_matrix_cooper_wallis_smallcases(28)
28 x 28 dense matrix over Integer Ring...

If existence is set to True, the function returns a boolean

sage: hadamard_matrix_cooper_wallis_smallcases(20, existence=True)
True

sage.combinat.matrices.hadamard_matrix.hadamard_matrix_from_sds(n, existence=False, check=True)

Construction of Hadamard matrices from supplementary difference sets.

Given four SDS with parameters \(4-\{n; n_1, n_2, n_3, n_4; \lambda\}\) with \(n_1 + n_2 + n_3 + n_4 = n+\lambda\) we can construct four (-1,+1) sequences \(a_i = (a_{i,0},...,a_{i,n-1})\) where \(a_{i,j} = -1\) iff \(j \in S_i\). These will be the fist rows of four circulant matrices \(A_1, A_2, A_3, A_4\) which, when plugged into the Goethals-Seidel array, create an Hadamard matrix of order \(4n\) (see [Djo1994b]).

The supplementary difference sets are taken from sage.combinat.designs.difference_family.supplementary_difference_set().

INPUT:

• n – integer; the order of the matrix to be constructed

• check – boolean (default: True); if True, check that the matrix is a Hadamard before returning

• existence – boolean (default: False); if True, only check if the matrix exists

OUTPUT:

If existence=False, returns the Hadamard matrix of order \(n\). It raises an error if no data is available to construct the matrix of the given order, or if \(n\) is not a multiple of \(4\). If existence=True, returns a boolean representing whether the matrix can be constructed or not.

EXAMPLES:

By default The function returns the Hadamard matrix

sage: from sage.combinat.matrices.hadamard_matrix import hadamard_matrix_from_sds
sage: hadamard_matrix_from_sds(148)
148 x 148 dense matrix over Integer Ring...

If existence is set to True, the function returns a boolean

sage: hadamard_matrix_from_sds(764, existence=True)
True

sage.combinat.matrices.hadamard_matrix.hadamard_matrix_miyamoto_construction(n, existence=False, check=True)

Construct Hadamard matrix using the Miyamoto construction.

If \(q = n/4\) is a prime power, and there exists a Hadamard matrix of order \(q-1\), then a Hadamard matrix of order \(n\) can be constructed (see [Miy1991]).

INPUT:

• n – integer; the order of the matrix to be constructed

• check – boolean (default: True); if True, check that the matrix is a Hadamard before returning

• existence – boolean (default: False); if True, only check if the matrix exists

OUTPUT:

If existence=False, returns the Hadamard matrix of order \(n\). It raises an error if no data is available to construct the matrix of the given order, or if \(n\) does not satisfies the constraints. If existence=True, returns a boolean representing whether the matrix can be constructed or not.

EXAMPLES:

By default the function returns the Hadamard matrix

sage: from sage.combinat.matrices.hadamard_matrix import hadamard_matrix_miyamoto_construction
sage: hadamard_matrix_miyamoto_construction(20)
20 x 20 dense matrix over Integer Ring...

If existence is set to True, the function returns a boolean

sage: hadamard_matrix_miyamoto_construction(36, existence=True)
True

sage.combinat.matrices.hadamard_matrix.hadamard_matrix_paleyI(n, normalize=True)

Implement the Paley type I construction.

The Paley type I case corresponds to the case \(p=n-1 \cong 3 \mod{4}\) for a prime power \(p\) (see [Hora]).

INPUT:

• n – the matrix size

• normalize– boolean (default: True); whether to normalize the result

EXAMPLES:

We note that this method by default returns a normalised Hadamard matrix

sage: from sage.combinat.matrices.hadamard_matrix import hadamard_matrix_paleyI
sage: hadamard_matrix_paleyI(4)
[ 1  1  1  1]
[ 1 -1  1 -1]
[ 1 -1 -1  1]
[ 1  1 -1 -1]

Otherwise, it returns a skew Hadamard matrix \(H\), i.e. \(H=S+I\), with \(S=-S^\top\)

sage: M = hadamard_matrix_paleyI(4, normalize=False); M
[ 1  1  1  1]
[-1  1  1 -1]
[-1 -1  1  1]
[-1  1 -1  1]
sage: S = M - identity_matrix(4); -S == S.T
True

sage.combinat.matrices.hadamard_matrix.hadamard_matrix_paleyII(n)

Implement the Paley type II construction.

The Paley type II case corresponds to the case \(p=n/2-1 \cong 1 \mod{4}\) for a prime power \(p\) (see [Hora]).

EXAMPLES:

sage: sage.combinat.matrices.hadamard_matrix.hadamard_matrix_paleyII(12).det()
2985984
sage: 12^6
2985984

We note that the method returns a normalised Hadamard matrix

sage: sage.combinat.matrices.hadamard_matrix.hadamard_matrix_paleyII(12)
[ 1  1| 1  1| 1  1| 1  1| 1  1| 1  1]
[ 1 -1|-1  1|-1  1|-1  1|-1  1|-1  1]
[-----+-----+-----+-----+-----+-----]
[ 1 -1| 1 -1| 1  1|-1 -1|-1 -1| 1  1]
[ 1  1|-1 -1| 1 -1|-1  1|-1  1| 1 -1]
[-----+-----+-----+-----+-----+-----]
[ 1 -1| 1  1| 1 -1| 1  1|-1 -1|-1 -1]
[ 1  1| 1 -1|-1 -1| 1 -1|-1  1|-1  1]
[-----+-----+-----+-----+-----+-----]
[ 1 -1|-1 -1| 1  1| 1 -1| 1  1|-1 -1]
[ 1  1|-1  1| 1 -1|-1 -1| 1 -1|-1  1]
[-----+-----+-----+-----+-----+-----]
[ 1 -1|-1 -1|-1 -1| 1  1| 1 -1| 1  1]
[ 1  1|-1  1|-1  1| 1 -1|-1 -1| 1 -1]
[-----+-----+-----+-----+-----+-----]
[ 1 -1| 1  1|-1 -1|-1 -1| 1  1| 1 -1]
[ 1  1| 1 -1|-1  1|-1  1| 1 -1|-1 -1]

sage.combinat.matrices.hadamard_matrix.hadamard_matrix_spence_construction(n, existence=False, check=True)

Create a Hadamard matrix of order \(n\) using the Spence construction.

This construction (detailed in [Spe1975]), uses supplementary difference sets implemented in sage.combinat.designs.difference_family.supplementary_difference_set_from_rel_diff_set() to create the desired matrix.

INPUT:

• n – integer; the order of the matrix to be constructed

• existence – boolean (default: False); if True, only check if the matrix exists

• check – bolean (default: True); if True, check that the matrix is a Hadamard matrix before returning

OUTPUT:

If existence=True, returns a boolean representing whether the Hadamard matrix can be constructed. Otherwise, returns the Hadamard matrix, or raises an error if it cannot be constructed.

EXAMPLES:

sage: from sage.combinat.matrices.hadamard_matrix import hadamard_matrix_spence_construction
sage: hadamard_matrix_spence_construction(36)
36 x 36 dense matrix over Integer Ring...

If existence is True, the function returns a boolean

sage: hadamard_matrix_spence_construction(52, existence=True)
True

sage.combinat.matrices.hadamard_matrix.hadamard_matrix_turyn_type(a, b, c, d, e1, e2, e3, e4, check=True)

Construction of Turyn type Hadamard matrix.

Given \(n\times n\) circulant matrices \(A\), \(B\), \(C\), \(D\) with 1,-1 entries, satisfying \(AA^\top + BB^\top + CC^\top + DD^\top = 4nI\), and a set of Baumert-Hall units of order \(4t\), one can construct a Hadamard matrix of order \(4tn\) as detailed by Turyn in [Tur1974].

INPUT:

• a – 1,-1 list; the 1st row of \(A\)

• b – 1,-1 list; the 1st row of \(B\)

• d – 1,-1 list; the 1st row of \(C\)

• c – 1,-1 list; the 1st row of \(D\)

• e1 – Matrix; the first Baumert-Hall unit

• e2 – Matrix; the second Baumert-Hall unit

• e3 – Matrix; the third Baumert-Hall unit

• e4 – Matrix; the fourth Baumert-Hall unit

• check – boolean (default: True); whether to check that the output is a Hadamard matrix before returning it

EXAMPLES:

sage: from sage.combinat.matrices.hadamard_matrix import hadamard_matrix_turyn_type, _get_baumert_hall_units
sage: A, B, C, D = _get_baumert_hall_units(28)
sage: hadamard_matrix_turyn_type([1], [1], [1], [1], A, B, C, D)
28 x 28 dense matrix over Integer Ring...

sage.combinat.matrices.hadamard_matrix.hadamard_matrix_williamson_type(a, b, c, d, check=True)

Construction of Williamson type Hadamard matrix.

Given \(n\times n\) circulant matrices \(A\), \(B\), \(C\), \(D\) with 1,-1 entries, and satisfying \(AA^\top + BB^\top + CC^\top + DD^\top = 4nI\), one can construct a Hadamard matrix of order \(4n\), cf. [Ha83].

INPUT:

• a – (1,-1) list; the 1st row of \(A\)

• b – (1,-1) list; the 1st row of \(B\)

• d – (1,-1) list; the 1st row of \(C\)

• c – (1,-1) list; the 1st row of \(D\)

• check – boolean (default: True); whether to check that the output is a Hadamard matrix before returning it

EXAMPLES:

sage: from sage.combinat.matrices.hadamard_matrix import hadamard_matrix_williamson_type
sage: a = [ 1,  1, 1]
sage: b = [ 1, -1, -1]
sage: c = [ 1, -1, -1]
sage: d = [ 1, -1, -1]
sage: M = hadamard_matrix_williamson_type(a,b,c,d,check=True)

sage.combinat.matrices.hadamard_matrix.hadamard_matrix_www(url_file, comments=False)

Pull file from Sloane’s database and return the corresponding Hadamard matrix as a Sage matrix.

You must input a filename of the form “had.n.xxx.txt” as described on the webpage http://neilsloane.com/hadamard/, where “xxx” could be empty or a number of some characters.

If comments=True then the “Automorphism…” line of the had.n.xxx.txt file is printed if it exists. Otherwise nothing is done.

EXAMPLES:

sage: hadamard_matrix_www("had.4.txt")      # optional - internet
[ 1  1  1  1]
[ 1 -1  1 -1]
[ 1  1 -1 -1]
[ 1 -1 -1  1]
sage: hadamard_matrix_www("had.16.2.txt",comments=True)   # optional - internet
Automorphism group has order = 49152 = 2^14 * 3
[ 1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1]
[ 1 -1  1 -1  1 -1  1 -1  1 -1  1 -1  1 -1  1 -1]
[ 1  1 -1 -1  1  1 -1 -1  1  1 -1 -1  1  1 -1 -1]
[ 1 -1 -1  1  1 -1 -1  1  1 -1 -1  1  1 -1 -1  1]
[ 1  1  1  1 -1 -1 -1 -1  1  1  1  1 -1 -1 -1 -1]
[ 1 -1  1 -1 -1  1 -1  1  1 -1  1 -1 -1  1 -1  1]
[ 1  1 -1 -1 -1 -1  1  1  1  1 -1 -1 -1 -1  1  1]
[ 1 -1 -1  1 -1  1  1 -1  1 -1 -1  1 -1  1  1 -1]
[ 1  1  1  1  1  1  1  1 -1 -1 -1 -1 -1 -1 -1 -1]
[ 1  1  1  1 -1 -1 -1 -1 -1 -1 -1 -1  1  1  1  1]
[ 1  1 -1 -1  1 -1  1 -1 -1 -1  1  1 -1  1 -1  1]
[ 1  1 -1 -1 -1  1 -1  1 -1 -1  1  1  1 -1  1 -1]
[ 1 -1  1 -1  1 -1 -1  1 -1  1 -1  1 -1  1  1 -1]
[ 1 -1  1 -1 -1  1  1 -1 -1  1 -1  1  1 -1 -1  1]
[ 1 -1 -1  1  1  1 -1 -1 -1  1  1 -1 -1 -1  1  1]
[ 1 -1 -1  1 -1 -1  1  1 -1  1  1 -1  1  1 -1 -1]

sage.combinat.matrices.hadamard_matrix.is_hadamard_matrix(M, normalized=False, skew=False, verbose=False)

Test if M is a Hadamard matrix.

INPUT:

• M – a matrix

• normalized – boolean (default: False); whether to test if M is a normalized Hadamard matrix, i.e. has its first row/column filled with +1

• skew – boolean (default: False); whether to test if M is a skew Hadamard matrix, i.e. \(M=S+I\) for \(-S=S^\top\), and \(I\) the identity matrix

• verbose – boolean (default: False); whether to be verbose when the matrix is not Hadamard

EXAMPLES:

sage: from sage.combinat.matrices.hadamard_matrix import is_hadamard_matrix
sage: h = matrix.hadamard(12)
sage: is_hadamard_matrix(h)
True
sage: from sage.combinat.matrices.hadamard_matrix import skew_hadamard_matrix
sage: h = skew_hadamard_matrix(12)
sage: is_hadamard_matrix(h, skew=True)
True
sage: h = matrix.hadamard(12)
sage: h[0,0] = 2
sage: is_hadamard_matrix(h,verbose=True)
The matrix does not only contain +1 and -1 entries, e.g. 2
False
sage: h = matrix.hadamard(12)
sage: for i in range(12):
....:     h[i,2] = -h[i,2]
sage: is_hadamard_matrix(h,verbose=True,normalized=True)
The matrix is not normalized
False

sage.combinat.matrices.hadamard_matrix.is_skew_hadamard_matrix(M, normalized=False, verbose=False)

Test if M is a skew Hadamard matrix.

this is a wrapper around the function is_hadamard_matrix()

INPUT:

• M – a matrix

• normalized – boolean (default: False); whether to test if M is a skew-normalized Hadamard matrix, i.e. has its first row filled with +1

• verbose – boolean (default: False); whether to be verbose when the matrix is not skew Hadamard

EXAMPLES:

sage: from sage.combinat.matrices.hadamard_matrix import is_skew_hadamard_matrix, skew_hadamard_matrix
sage: h = matrix.hadamard(12)
sage: is_skew_hadamard_matrix(h, verbose=True)
The matrix is not skew
False
sage: h = skew_hadamard_matrix(12)
sage: is_skew_hadamard_matrix(h)
True
sage: from sage.combinat.matrices.hadamard_matrix import normalise_hadamard
sage: h = normalise_hadamard(skew_hadamard_matrix(12), skew=True)
sage: is_skew_hadamard_matrix(h, verbose=True, normalized=True)
True

sage.combinat.matrices.hadamard_matrix.normalise_hadamard(H, skew=False)

Return the normalised Hadamard matrix corresponding to H.

The normalised Hadamard matrix corresponding to a Hadamard matrix \(H\) is a matrix whose every entry in the first row and column is +1.

If skew is True, the matrix returned will be skew-normal: a skew Hadamard matrix with first row of all \(+1\).

EXAMPLES:

sage: from sage.combinat.matrices.hadamard_matrix import normalise_hadamard, is_hadamard_matrix, skew_hadamard_matrix
sage: H = normalise_hadamard(hadamard_matrix(4))
sage: H == hadamard_matrix(4)
True
sage: H = normalise_hadamard(skew_hadamard_matrix(20, skew_normalize=False), skew=True)
sage: is_hadamard_matrix(H, skew=True, normalized=True)
True

If skew is True but the Hadamard matrix is not skew, the matrix returned will not be normalized:

sage: H = normalise_hadamard(hadamard_matrix(92), skew=True)
sage: is_hadamard_matrix(H, normalized=True)
False

sage.combinat.matrices.hadamard_matrix.regular_symmetric_hadamard_matrix_with_constant_diagonal(n, e, existence=False)

Return a Regular Symmetric Hadamard Matrix with Constant Diagonal.

A Hadamard matrix is said to be regular if its rows all sum to the same value.

For \(\epsilon\in\{-1,+1\}\), we say that \(M\) is a \((n,\epsilon)-RSHCD\) if \(M\) is a regular symmetric Hadamard matrix with constant diagonal \(\delta\in\{-1,+1\}\) and row sums all equal to \(\delta \epsilon \sqrt(n)\). For more information, see [HX2010] or 10.5.1 in [BH2012]. For the case \(n=324\), see RSHCD_324() and [CP2016].

INPUT:

• n – integer; side of the matrix

• e – \(-1\) or \(+1\); the value of \(\epsilon\)

EXAMPLES:

sage: from sage.combinat.matrices.hadamard_matrix import regular_symmetric_hadamard_matrix_with_constant_diagonal
sage: regular_symmetric_hadamard_matrix_with_constant_diagonal(4,1)
[ 1  1  1 -1]
[ 1  1 -1  1]
[ 1 -1  1  1]
[-1  1  1  1]
sage: regular_symmetric_hadamard_matrix_with_constant_diagonal(4,-1)
[ 1 -1 -1 -1]
[-1  1 -1 -1]
[-1 -1  1 -1]
[-1 -1 -1  1]

Other hardcoded values:

sage: for n,e in [(36,1),(36,-1),(100,1),(100,-1),(196, 1)]:  # long time
....:     print(repr(regular_symmetric_hadamard_matrix_with_constant_diagonal(n,e)))
36 x 36 dense matrix over Integer Ring
36 x 36 dense matrix over Integer Ring
100 x 100 dense matrix over Integer Ring
100 x 100 dense matrix over Integer Ring
196 x 196 dense matrix over Integer Ring

sage: for n,e in [(324,1),(324,-1)]: # not tested - long time, tested in RSHCD_324
....:     print(repr(regular_symmetric_hadamard_matrix_with_constant_diagonal(n,e)))
324 x 324 dense matrix over Integer Ring
324 x 324 dense matrix over Integer Ring

From two close prime powers:

sage: regular_symmetric_hadamard_matrix_with_constant_diagonal(64,-1)
64 x 64 dense matrix over Integer Ring (use the '.str()' method to see the entries)

From a prime power and a conference matrix:

sage: regular_symmetric_hadamard_matrix_with_constant_diagonal(676,1)  # long time
676 x 676 dense matrix over Integer Ring (use the '.str()' method to see the entries)

Recursive construction:

sage: regular_symmetric_hadamard_matrix_with_constant_diagonal(144,-1)
144 x 144 dense matrix over Integer Ring (use the '.str()' method to see the entries)

REFERENCE:

• [BH2012]

• [HX2010]

sage.combinat.matrices.hadamard_matrix.rshcd_from_close_prime_powers(n)

Return a \((n^2,1)\)-RSHCD when \(n-1\) and \(n+1\) are odd prime powers and \(n=0\pmod{4}\).

The construction implemented here appears in Theorem 4.3 from [GS1970].

Note that the authors of [SWW1972] claim in Corollary 5.12 (page 342) to have proved the same result without the \(n=0\pmod{4}\) restriction with a very similar construction. So far, however, I (Nathann Cohen) have not been able to make it work.

INPUT:

• n – an integer congruent to \(0\pmod{4}\)

See also

regular_symmetric_hadamard_matrix_with_constant_diagonal()

EXAMPLES:

sage: from sage.combinat.matrices.hadamard_matrix import rshcd_from_close_prime_powers
sage: rshcd_from_close_prime_powers(4)
[-1 -1  1 -1  1 -1 -1  1 -1  1 -1 -1  1 -1  1 -1]
[-1 -1 -1  1  1 -1 -1  1 -1 -1  1 -1 -1  1 -1  1]
[ 1 -1 -1  1  1  1  1 -1 -1 -1 -1 -1 -1 -1  1 -1]
[-1  1  1 -1  1  1 -1 -1 -1 -1 -1  1 -1 -1 -1  1]
[ 1  1  1  1 -1 -1 -1 -1 -1 -1  1 -1  1 -1 -1 -1]
[-1 -1  1  1 -1 -1  1 -1 -1  1 -1  1 -1  1 -1 -1]
[-1 -1  1 -1 -1  1 -1 -1  1 -1  1 -1 -1  1  1 -1]
[ 1  1 -1 -1 -1 -1 -1 -1 -1  1 -1 -1 -1  1  1  1]
[-1 -1 -1 -1 -1 -1  1 -1 -1 -1  1  1  1 -1  1  1]
[ 1 -1 -1 -1 -1  1 -1  1 -1 -1 -1  1  1  1 -1 -1]
[-1  1 -1 -1  1 -1  1 -1  1 -1 -1 -1  1  1 -1 -1]
[-1 -1 -1  1 -1  1 -1 -1  1  1 -1 -1  1 -1 -1  1]
[ 1 -1 -1 -1  1 -1 -1 -1  1  1  1  1 -1 -1 -1 -1]
[-1  1 -1 -1 -1  1  1  1 -1  1  1 -1 -1 -1 -1 -1]
[ 1 -1  1 -1 -1 -1  1  1  1 -1 -1 -1 -1 -1 -1  1]
[-1  1 -1  1 -1 -1 -1  1  1 -1 -1  1 -1 -1  1 -1]

REFERENCE:

• [SWW1972]

sage.combinat.matrices.hadamard_matrix.rshcd_from_prime_power_and_conference_matrix(n)

Return a \(((n-1)^2,1)\)-RSHCD if \(n\) is prime power, and symmetric \((n-1)\)-conference matrix exists

The construction implemented here is Theorem 16 (and Corollary 17) from [WW1972].

In [SWW1972] this construction (Theorem 5.15 and Corollary 5.16) is reproduced with a typo. Note that [WW1972] refers to [Sz1969] for the construction, provided by szekeres_difference_set_pair(), of complementary difference sets, and the latter has a typo.

From a symmetric_conference_matrix(), we only need the Seidel adjacency matrix of the underlying strongly regular conference (i.e. Paley type) graph, which we construct directly.

INPUT:

• n – an integer

See also

regular_symmetric_hadamard_matrix_with_constant_diagonal()

EXAMPLES:

A 36x36 example

sage: from sage.combinat.matrices.hadamard_matrix import rshcd_from_prime_power_and_conference_matrix
sage: from sage.combinat.matrices.hadamard_matrix import is_hadamard_matrix
sage: H = rshcd_from_prime_power_and_conference_matrix(7); H
36 x 36 dense matrix over Integer Ring (use the '.str()' method to see the entries)
sage: H == H.T and is_hadamard_matrix(H) and H.diagonal() == [1]*36 and list(sum(H)) == [6]*36
True

Bigger examples, only provided by this construction

sage: H = rshcd_from_prime_power_and_conference_matrix(27)  # long time
sage: H == H.T and is_hadamard_matrix(H)                    # long time
True
sage: H.diagonal() == [1]*676 and list(sum(H)) == [26]*676  # long time
True

In this example the conference matrix is not Paley, as 45 is not a prime power

sage: H = rshcd_from_prime_power_and_conference_matrix(47)  # not tested (long time)

REFERENCE:

• [WW1972]

sage.combinat.matrices.hadamard_matrix.skew_hadamard_matrix(n, existence=False, skew_normalize=True, check=True)

Tries to construct a skew Hadamard matrix.

A Hadamard matrix \(H\) is called skew if \(H=S-I\), for \(I\) the identity matrix and \(-S=S^\top\). Currently all orders \(\le 1200\) for which a construction is known are implemented. For \(n > 1200\), only some orders are available.

INPUT:

• n – integer; dimension of the matrix

• existence – boolean (default: False); whether to build the matrix or merely query if a construction is available in Sage. When set to True, the function returns:

  • True – meaning that Sage knows how to build the matrix

  • Unknown – meaning that Sage does not know how to build the matrix, but that the design may exist (see sage.misc.unknown).

  • False – meaning that the matrix does not exist.

• skew_normalize – boolean (default: True); whether to make the 1st row all-one, and adjust the 1st column accordingly

• check – boolean (default: True); whether to check that output is correct before returning it. As this is expected to be useless (but we are cautious guys), you may want to disable it whenever you want speed

EXAMPLES:

sage: from sage.combinat.matrices.hadamard_matrix import skew_hadamard_matrix
sage: skew_hadamard_matrix(12).det()
2985984
sage: 12^6
2985984
sage: skew_hadamard_matrix(1)
[1]
sage: skew_hadamard_matrix(2)
[ 1  1]
[-1  1]

REFERENCES:

• [Ha83]

sage.combinat.matrices.hadamard_matrix.skew_hadamard_matrix_from_complementary_difference_sets(n, existence=False, check=True)

Construct a skew Hadamard matrix of order \(n=4(m+1)\) from complementary difference sets.

If \(A, B\) are complementary difference sets over a group of order \(2m+1\), then they can be used to construct a skew Hadamard matrix, as described in [BS1969].

The complementary difference sets are constructed using the function sage.combinat.designs.difference_family.complementary_difference_sets().

INPUT:

• n – positive integer; the order of the matrix to be constructed

• existence – boolean (default: False); if True, only return whether the skew Hadamard matrix can be constructed

• check – boolean (default: True); if True, check that the result is a skew Hadamard matrix before returning it

OUTPUT:

If existence=False, returns the skew Hadamard matrix of order \(n\). It raises an error if \(n\) does not satisfy the required conditions. If existence=True, returns a boolean representing whether the matrix can be constructed or not.

EXAMPLES:

sage: from sage.combinat.matrices.hadamard_matrix import skew_hadamard_matrix_from_complementary_difference_sets
sage: skew_hadamard_matrix_from_complementary_difference_sets(20)
20 x 20 dense matrix over Integer Ring...
sage: skew_hadamard_matrix_from_complementary_difference_sets(52, existence=True)
True

sage.combinat.matrices.hadamard_matrix.skew_hadamard_matrix_from_good_matrices(a, b, c, d, check=True)

Construct skew Hadamard matrix from good matrices.

Given good matrices \(A\), \(B\), \(C\), \(D\) (\(A\) circulant, \(B, C, D\) back-circulant) they can be used to construct a skew Hadamard matrix using the following block matrix (as described in [Sze1988]):

\[\begin{split}\left(\begin{array}{rrrr} A & B & C & D \\ -B & A & D & -C \\ -C & -D & A & B \\ -D & C & -B & A \end{array}\right)\end{split}\]

INPUT:

• a – (1,-1) list; the 1st row of \(A\)

• b – (1,-1) list; the 1st row of \(B\)

• d – (1,-1) list; the 1st row of \(C\)

• c – (1,-1) list; the 1st row of \(D\)

• check – boolean (default: True); if True, check that the matrix is a skew Hadamard matrix before returning it

EXAMPLES:

sage: from sage.combinat.matrices.hadamard_matrix import skew_hadamard_matrix_from_good_matrices
sage: a, b, c, d = ([1, 1, 1, -1, -1], [1, -1, 1, 1, -1], [1, -1, -1, -1, -1], [1, -1, -1, -1, -1])
sage: skew_hadamard_matrix_from_good_matrices(a, b, c, d)
20 x 20 dense matrix over Integer Ring...

sage.combinat.matrices.hadamard_matrix.skew_hadamard_matrix_from_good_matrices_smallcases(n, existence=False, check=True)

Construct skew Hadamard matrices from good matrices for some small values of \(n=4m\), with \(m\) odd.

The function stores good matrices of odd orders \(\le 31\), taken from [Sze1988]. These are used to create skew Hadamard matrices of order \(4m\), \(1 \le m \le 31\) (\(m\) odd), using the function skew_hadamard_matrix_from_good_matrices().

ALGORITHM:

Given four sequences (stored in E_sequences) of length \(m\), they can be used to construct four \(E-sequences\) of length \(n=2m+1\), as follows:

\[\begin{split}\begin{aligned} a &= 1, a_0, a_1, ..., a_m, -a_m, -a_{m-1}, ..., -a_0 \\ b &= 1, b_0, b_1, ..., b_m, b_m, b_{m-1}, ..., b_0 \\ c &= 1, c_0, c_1, ..., c_m, c_m, c_{m-1}, ..., c_0 \\ d &= 1, d_0, d_1, ..., d_m, d_m, d_{m-1}, ..., d_0 \\ \end{aligned}\end{split}\]

These E-sequences will be the first rows of the four good matrices needed to construct a skew Hadamard matrix of order \(4n\).

INPUT:

• n – integer; the order of the skew Hadamard matrix to be constructed

• existence – boolean (default: False); If True, only return whether the Hadamard matrix can be constructed

• check – boolean (default: True): if True, check that the matrix is a Hadamard matrix before returning it

OUTPUT:

If existence=False, returns the skew Hadamard matrix of order \(n\). It raises an error if no data is available to construct the matrix of the given order. If existence=True, returns a boolean representing whether the matrix can be constructed or not.

EXAMPLES:

sage: from sage.combinat.matrices.hadamard_matrix import skew_hadamard_matrix_from_good_matrices_smallcases
sage: skew_hadamard_matrix_from_good_matrices_smallcases(20)
20 x 20 dense matrix over Integer Ring...
sage: skew_hadamard_matrix_from_good_matrices_smallcases(20, existence=True)
True

sage.combinat.matrices.hadamard_matrix.skew_hadamard_matrix_from_orthogonal_design(n, existence=False, check=True)

Construct skew Hadamard matrices of order \(mn(n - 1)\) if suitable orthogonal designs exist.

In [Seb1978] is proved that if amicable Hadamard matrices of order \(n\) and an orthogonal design of type \((1, m, mn - m - 1)\) in order \(mn\) exist, then a skew Hadamard matrix of order \(mn(n - 1)\) can be constructed. The paper uses amicable orthogonal designs instead of amicable Hadamard matrices, but the two are equivalent (see [Seb2017]).

Amicable Hadamard matrices are constructed using amicable_hadamard_matrices(), and the orthogonal designs are constructed using the Goethals-Seidel array, with data taken from [Seb2017].

INPUT:

• n – positive integer; the order of the matrix to be constructed

• existence – boolean (default: False); if True, only return whether the skew Hadamard matrix can be constructed

• check – boolean (default: True); if True, check that the result is a skew Hadamard matrix before returning it

OUTPUT:

If existence=False, returns the skew Hadamard matrix of order \(n\). It raises an error if a construction for order \(n\) is not yet implemented, or if \(n\) does not satisfy the constraint. If existence=True, returns a boolean representing whether the matrix can be constructed or not.

EXAMPLES:

sage: from sage.combinat.matrices.hadamard_matrix import skew_hadamard_matrix_from_orthogonal_design
sage: skew_hadamard_matrix_from_orthogonal_design(756)
756 x 756 dense matrix over Integer Ring...

If existence is True, the function returns a boolean:

sage: skew_hadamard_matrix_from_orthogonal_design(200, existence=True)
False

sage.combinat.matrices.hadamard_matrix.skew_hadamard_matrix_spence_1975(n, existence=False, check=True)

Construct a skew Hadamard matrix of order \(n = 4(1 + q + q^2)\) using the Spence construction.

If \(n = 4(1 + q + q^2)\) where \(q\) is a prime power such that either \(1 + q + q^2\) is a prime congruent to \(3, 5, 7 \mod 8\) or \(3 + 2q + 2q^2\) is a prime power, then a skew Hadamard matrix of order \(n\) can be constructed using the Goethals Seidel array. The four matrices \(A, B, C, D\) plugged into the GS-array are created using complementary difference sets of order \(1 + q + q^2\) (which exist if \(q\) satisfies the conditions above), and a cyclic planar difference set with parameters \((1 + q^2 + q^4, 1 + q^2, 1)\). These are constructed by the functions sage.combinat.designs.difference_family.complementary_difference_sets() and sage.combinat.designs.difference_family.difference_family().

For more details, see [Spe1975b].

INPUT:

• n – positive integer; the order of the matrix to be constructed

• existence – boolean (default: False); if True, only return whether the Hadamard matrix can be constructed

• check – boolean (default: True); check that the result is a skew Hadamard matrix before returning it

OUTPUT:

If existence=False, returns the skew Hadamard matrix of order \(n\). It raises an error if \(n\) does not satisfy the required conditions. If existence=True, returns a boolean representing whether the matrix can be constructed or not.

EXAMPLES:

sage: from sage.combinat.matrices.hadamard_matrix import skew_hadamard_matrix_spence_1975
sage: skew_hadamard_matrix_spence_1975(52)
52 x 52 dense matrix over Integer Ring...

If existence is True, the function returns a boolean:

sage: skew_hadamard_matrix_spence_1975(52, existence=True)
True

sage.combinat.matrices.hadamard_matrix.skew_hadamard_matrix_spence_construction(n, check=True)

Construct skew Hadamard matrix of order \(n\) using Spence constrution.

This function will construct skew Hadamard matrix of order \(n=2(q+1)\) where \(q\) is a prime power with \(q = 5\) (mod 8). The construction is taken from [Spe1977], and the relative difference sets are constructed using sage.combinat.designs.difference_family.relative_difference_set_from_homomorphism().

INPUT:

• n – positive integer

• check – boolean (default: True); if True, check that the resulting matrix is Hadamard before returning it

OUTPUT:

If n satisfies the requirements described above, the function returns a \(n\times n\) Hadamard matrix. Otherwise, an exception is raised.

EXAMPLES:

sage: from sage.combinat.matrices.hadamard_matrix import skew_hadamard_matrix_spence_construction
sage: skew_hadamard_matrix_spence_construction(28)
28 x 28 dense matrix over Integer Ring...

sage.combinat.matrices.hadamard_matrix.skew_hadamard_matrix_whiteman_construction(n, existence=False, check=True)

Construct a skew Hadamard matrix of order \(n=2(q+1)\) where \(q=p^t\) is a prime power with \(p \cong 5 \mod 8\) and \(t \cong 2 \mod 4\).

Assuming \(n\) satisfies the conditions above, it is possible to construct two supplementary difference sets \(A, B\) (see [Whi1971]), and these can be used to construct a skew Hadamard matrix, as described in [BS1969].

INPUT:

• n – positive integer; the order of the matrix to be constructed

• existence – boolean (default: False); If True, only return whether the Hadamard matrix can be constructed

• check – boolean (default: True); if True, check that the result is a skew Hadamard matrix before returning it

OUTPUT:

If existence=False, returns the skew Hadamard matrix of order \(n\). It raises an error if \(n\) does not satisfy the required conditions. If existence=True, returns a boolean representing whether the matrix can be constructed or not.

EXAMPLES:

sage: from sage.combinat.matrices.hadamard_matrix import skew_hadamard_matrix_whiteman_construction
sage: skew_hadamard_matrix_whiteman_construction(52)
52 x 52 dense matrix over Integer Ring...
sage: skew_hadamard_matrix_whiteman_construction(52, existence=True)
True

Note

A more general version of this construction is skew_hadamard_matrix_from_complementary_difference_sets().

sage.combinat.matrices.hadamard_matrix.symmetric_conference_matrix(n, check=True)

Tries to construct a symmetric conference matrix

A conference matrix is an \(n\times n\) matrix \(C\) with 0s on the main diagonal and 1s and -1s elsewhere, satisfying \(CC^\top=(n-1)I\). If \(C=C^\top\) then \(n \cong 2 \mod 4\) and \(C\) is Seidel adjacency matrix of a graph, whose descendent graphs are strongly regular graphs with parameters \((n-1,(n-2)/2,(n-6)/4,(n-2)/4)\), see Sec.10.4 of [BH2012]. Thus we build \(C\) from the Seidel adjacency matrix of the latter by adding row and column of 1s.

INPUT:

• n – integer; dimension of the matrix

• check – boolean (default: True); whether to check that output is correct before returning it. As this is expected to be useless (but we are cautious guys), you may want to disable it whenever you want speed

EXAMPLES:

sage: from sage.combinat.matrices.hadamard_matrix import symmetric_conference_matrix
sage: C = symmetric_conference_matrix(10); C
[ 0  1  1  1  1  1  1  1  1  1]
[ 1  0 -1 -1  1 -1  1  1  1 -1]
[ 1 -1  0 -1  1  1 -1 -1  1  1]
[ 1 -1 -1  0 -1  1  1  1 -1  1]
[ 1  1  1 -1  0 -1 -1  1 -1  1]
[ 1 -1  1  1 -1  0 -1  1  1 -1]
[ 1  1 -1  1 -1 -1  0 -1  1  1]
[ 1  1 -1  1  1  1 -1  0 -1 -1]
[ 1  1  1 -1 -1  1  1 -1  0 -1]
[ 1 -1  1  1  1 -1  1 -1 -1  0]
sage: C^2 == 9*identity_matrix(10) and C == C.T
True

sage.combinat.matrices.hadamard_matrix.symmetric_conference_matrix_paley(n)

Construct a symmetric conference matrix of order n.

A conference matrix is an \(n\times n\) matrix \(C\) with 0s on the main diagonal and 1s and -1s elsewhere, satisfying \(CC^\top=(n-1)I\). This construction assumes that \(q = n-1\) is a prime power, with \(q \cong 1 \mod 4\). See [Hora] or [Lon2013].

These matrices are used in hadamard_matrix_paleyII().

INPUT:

• n – integer; the order of the symmetric conference matrix to construct

EXAMPLES:

sage: from sage.combinat.matrices.hadamard_matrix import symmetric_conference_matrix_paley
sage: symmetric_conference_matrix_paley(6)
[ 0  1  1  1  1  1]
[ 1  0  1 -1 -1  1]
[ 1  1  0  1 -1 -1]
[ 1 -1  1  0  1 -1]
[ 1 -1 -1  1  0  1]
[ 1  1 -1 -1  1  0]

sage.combinat.matrices.hadamard_matrix.szekeres_difference_set_pair(m, check=True)

Construct Szekeres \((2m+1,m,1)\)-cyclic difference family

Let \(4m+3\) be a prime power. Theorem 3 in [Sz1969] contains a construction of a pair of complementary difference sets \(A\), \(B\) in the subgroup \(G\) of the quadratic residues in \(F_{4m+3}^*\). Namely \(|A|=|B|=m\), \(a\in A\) whenever \(a-1\in G\), \(b\in B\) whenever \(b+1 \in G\). See also Theorem 2.6 in [SWW1972] (there the formula for \(B\) is correct, as opposed to (4.2) in [Sz1969], where the sign before \(1\) is wrong.

In modern terminology, for \(m>1\) the sets \(A\) and \(B\) form a difference family with parameters \((2m+1,m,1)\). I.e. each non-identity \(g \in G\) can be expressed uniquely as \(xy^{-1}\) for \(x,y \in A\) or \(x,y \in B\). Other, specific to this construction, properties of \(A\) and \(B\) are: for \(a\) in \(A\) one has \(a^{-1}\) not in \(A\), whereas for \(b\) in \(B\) one has \(b^{-1}\) in \(B\).

INPUT:

• m – integer; dimension of the matrix

• check – boolean (default: True); whether to check \(A\) and \(B\) for correctness

EXAMPLES:

sage: from sage.combinat.matrices.hadamard_matrix import szekeres_difference_set_pair
sage: G,A,B=szekeres_difference_set_pair(6)
sage: G,A,B=szekeres_difference_set_pair(7)

REFERENCE:

• [Sz1969]

sage.combinat.matrices.hadamard_matrix.turyn_type_hadamard_matrix_smallcases(n, existence=False, check=True)

Construct a Hadamard matrix of order \(n\) from available 4-symbol \(\delta\) codes and Williamson quadruples.

The function looks for Baumert-Hall units and Williamson type matrices from four_symbol_delta_code_smallcases() and williamson_type_quadruples_smallcases() and use them to construct a Hadamard matrix with the Turyn construction defined in hadamard_matrix_turyn_type().

INPUT:

• n – integer; the order of the matrix to be constructed

• existence – boolean (default: False): if True, only check if the matrix exists

• check – boolean (default: True): if True, check that the matrix is a Hadamard matrix before returning

EXAMPLES:

sage: from sage.combinat.matrices.hadamard_matrix import turyn_type_hadamard_matrix_smallcases
sage: turyn_type_hadamard_matrix_smallcases(28, existence=True)
True
sage: turyn_type_hadamard_matrix_smallcases(28)
28 x 28 dense matrix over Integer Ring...

sage.combinat.matrices.hadamard_matrix.typeI_matrix_difference_set(G, A)

(1,-1)-incidence type I matrix of a difference set \(A\) in \(G\)

Let \(A\) be a difference set in a group \(G\) of order \(n\). Return \(n\times n\) matrix \(M\) with \(M_{ij}=1\) if \(A_i A_j^{-1} \in A\), and \(M_{ij}=-1\) otherwise.

EXAMPLES:

sage: from sage.combinat.matrices.hadamard_matrix import szekeres_difference_set_pair
sage: from sage.combinat.matrices.hadamard_matrix import typeI_matrix_difference_set
sage: G,A,B=szekeres_difference_set_pair(2)
sage: typeI_matrix_difference_set(G,A)
[-1  1 -1 -1  1]
[-1 -1 -1  1  1]
[ 1  1 -1 -1 -1]
[ 1 -1  1 -1 -1]
[-1 -1  1  1 -1]

sage.combinat.matrices.hadamard_matrix.williamson_goethals_seidel_skew_hadamard_matrix(a, b, c, d, check=True)

Williamson-Goethals-Seidel construction of a skew Hadamard matrix

Given \(n\times n\) (anti)circulant matrices \(A\), \(B\), \(C\), \(D\) with 1,-1 entries, and satisfying \(A+A^\top = 2I\), \(AA^\top + BB^\top + CC^\top + DD^\top = 4nI\), one can construct a skew Hadamard matrix of order \(4n\), cf. [GS70s].

INPUT:

• a – 1,-1 list; the 1st row of \(A\)

• b – 1,-1 list; the 1st row of \(B\)

• d – 1,-1 list; the 1st row of \(C\)

• c – 1,-1 list; the 1st row of \(D\)

• check – boolean (default: True); if True, check that the resulting matrix is skew Hadamard before returning it

EXAMPLES:

sage: from sage.combinat.matrices.hadamard_matrix import williamson_goethals_seidel_skew_hadamard_matrix as WGS
sage: a = [ 1,  1, 1, -1,  1, -1,  1, -1, -1]
sage: b = [ 1, -1, 1,  1, -1, -1,  1,  1, -1]
sage: c = [-1, -1]+[1]*6+[-1]
sage: d = [ 1,  1, 1, -1,  1,  1, -1,  1,  1]
sage: M = WGS(a,b,c,d,check=True)

REFERENCES:

• [GS70s]

• [Wall71]

• [KoSt08]

sage.combinat.matrices.hadamard_matrix.williamson_hadamard_matrix_smallcases(n, existence=False, check=True)

Construct Williamson type Hadamard matrices for some small values of n.

This function uses the data contained in sage.combinat.matrices.hadamard_matrix.williamson_type_quadruples_smallcases() to create Hadamard matrices of the Williamson type, using the construction from sage.combinat.matrices.hadamard_matrix.hadamard_matrix_williamson_type().

INPUT:

• n – integer; the order of the matrix

• existence – boolean (dafault: False); if True, only check that we can do the construction

• check – boolean (default: True); if True check the result

sage.combinat.matrices.hadamard_matrix.williamson_type_quadruples_smallcases(n, existence=False)

Quadruples of matrices that can be used to construct Williamson type Hadamard matrices.

This function contains for some values of n, four \(n\times n\) matrices used in the Williamson construction of Hadamard matrices. Namely, the function returns the first row of 4 \(n\times n\) circulant matrices with the properties described in sage.combinat.matrices.hadamard_matrix.hadamard_matrix_williamson_type(). The matrices for \(n = 3, 5, ..., 29, 37, 43\) are given in [Ha83]. The matrices for \(n = 31, 33, 39, 41, 45, 49, 51, 55, 57, 61, 63\) are given in [Lon2013].

INPUT:

• n – integer; the order of the matrices to be returned

• existence – boolean (dafault: False); if True, only check that we have the quadruple

OUTPUT:

If existence is false, returns a tuple containing four vectors, each being the first line of one of the four matrices. It raises an error if no such matrices are available. If existence is true, returns a boolean representing whether the matrices are available or not.

EXAMPLES:

sage: from sage.combinat.matrices.hadamard_matrix import williamson_type_quadruples_smallcases
sage: williamson_type_quadruples_smallcases(29)
((1, 1, 1, -1, -1, -1, 1, 1, -1, -1, 1, -1, 1, -1, -1, -1, -1, 1, -1, 1, -1, -1, 1, 1, -1, -1, -1, 1, 1),
 (1, -1, 1, -1, -1, -1, 1, 1, -1, -1, 1, -1, 1, 1, 1, 1, 1, 1, -1, 1, -1, -1, 1, 1, -1, -1, -1, 1, -1),
 (1, 1, 1, 1, -1, 1, 1, -1, 1, -1, -1, -1, 1, 1, 1, 1, 1, 1, -1, -1, -1, 1, -1, 1, 1, -1, 1, 1, 1),
 (1, 1, -1, -1, 1, -1, -1, 1, -1, 1, 1, 1, -1, 1, 1, 1, 1, -1, 1, 1, 1, -1, 1, -1, -1, 1, -1, -1, 1))
sage: williamson_type_quadruples_smallcases(43, existence=True)
True