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r""" 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
.. MATH::
\left(\begin{array}{rr} 1 & 1 \\ 1 & -1 \end{array}\right)
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:
(a) `|det(H)|=n^{n/2}` or
(b) `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 the Paley constructions (see for example [Hora]_) and the Sylvester construction. It also allows you to pull a Hadamard matrix from the database at [HadaSloa]_.
AUTHORS:
- David Joyner (2009-05-17): initial version
REFERENCES:
.. [HadaSloa] \N.J.A. Sloane's Library of Hadamard Matrices, at http://neilsloane.com/hadamard/ .. [HadaWiki] Hadamard matrices on Wikipedia, :wikipedia:`Hadamard_matrix` .. [Hora] \K. J. Horadam, Hadamard Matrices and Their Applications, Princeton University Press, 2006. """
#***************************************************************************** # This program is free software: you can redistribute it and/or modify # it under the terms of the GNU General Public License as published by # the Free Software Foundation, either version 2 of the License, or # (at your option) any later version. # http://www.gnu.org/licenses/ #***************************************************************************** from __future__ import print_function
from six.moves import range from six import itervalues from six.moves.urllib.request import urlopen
from sage.rings.integer_ring import ZZ from sage.matrix.constructor import matrix, block_matrix, block_diagonal_matrix, diagonal_matrix from sage.arith.all import is_square, is_prime_power, divisors from math import sqrt from sage.matrix.constructor import identity_matrix as I from sage.matrix.constructor import ones_matrix as J from sage.misc.unknown import Unknown
def normalise_hadamard(H): """ 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.
EXAMPLES::
sage: H = sage.combinat.matrices.hadamard_matrix.normalise_hadamard(hadamard_matrix(4)) sage: H == hadamard_matrix(4) True """
def hadamard_matrix_paleyI(n, normalize=True): """ Implements the Paley type I construction.
The Paley type I case corresponds to the case `p \cong 3 \mod{4}` for a prime `p` (see [Hora]_).
INPUT:
- ``n`` -- the matrix size
- ``normalize`` (boolean) -- 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
TESTS::
sage: from sage.combinat.matrices.hadamard_matrix import is_hadamard_matrix sage: test_cases = [x+1 for x in range(100) if is_prime_power(x) and x%4==3] sage: all(is_hadamard_matrix(hadamard_matrix_paleyI(n),normalized=True,verbose=True) ....: for n in test_cases) True sage: all(is_hadamard_matrix(hadamard_matrix_paleyI(n,normalize=False),verbose=True) ....: for n in test_cases) True """ raise ValueError("The order %s is not covered by the Paley type I construction." % n)
for x in K_list] for y in K_list])
def hadamard_matrix_paleyII(n): """ Implements the Paley type II construction.
The Paley type II case corresponds to the case `p \cong 1 \mod{4}` for a prime `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]
TESTS::
sage: from sage.combinat.matrices.hadamard_matrix import (hadamard_matrix_paleyII, is_hadamard_matrix) sage: test_cases = [2*(x+1) for x in range(50) if is_prime_power(x) and x%4==1] sage: all(is_hadamard_matrix(hadamard_matrix_paleyII(n),normalized=True,verbose=True) ....: for n in test_cases) True """ raise ValueError("The order %s is not covered by the Paley type II construction." % n)
for x in K_list] for y in K_list])
1: matrix(2,2,[ 1, 1, 1,-1]), -1: matrix(2,2,[-1,-1,-1, 1])}
def is_hadamard_matrix(M, normalized=False, skew=False, verbose=False): r""" Test if `M` is a hadamard matrix.
INPUT:
- ``M`` -- a matrix
- ``normalized`` (boolean) -- whether to test if ``M`` is a normalized Hadamard matrix, i.e. has its first row/column filled with +1.
- ``skew`` (boolean) -- 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) -- 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
TESTS::
sage: h = matrix.hadamard(12) sage: is_hadamard_matrix(h, skew=True) False sage: is_hadamard_matrix(h, skew=True, verbose=True) The matrix is not skew False sage: h=skew_hadamard_matrix(12) sage: is_hadamard_matrix(h, skew=True, verbose=True) True sage: is_hadamard_matrix(h, skew=False, verbose=True) True sage: h=-h sage: is_hadamard_matrix(h, skew=True, verbose=True) The matrix is not skew - diagonal entries must be all 1 False sage: is_hadamard_matrix(h, skew=False, verbose=True) True """ if verbose: print("The matrix is not square ({}x{})".format(M.nrows(), n)) return False
return True
set(itervalues(prod)) != {n} or any((i, i) not in prod for i in range(n))): if verbose: print("The product M*M.transpose() is not equal to nI") return False
set(M.column(0)) != {1}):
from sage.matrix.constructor import matrix_method @matrix_method def hadamard_matrix(n,existence=False, check=True): r""" Tries to construct a Hadamard matrix using a combination of Paley and Sylvester constructions.
INPUT:
- ``n`` (integer) -- dimension of the matrix
- ``existence`` (boolean) -- 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 :mod:`sage.misc.unknown`).
- ``False`` -- meaning that the matrix does not exist.
- ``check`` (boolean) -- 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. Set to ``True`` by default.
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 :func:`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]
TESTS::
sage: matrix.hadamard(10,existence=True) False sage: matrix.hadamard(12,existence=True) True sage: matrix.hadamard(92,existence=True) Unknown sage: matrix.hadamard(10) Traceback (most recent call last): ... ValueError: The Hadamard matrix of order 10 does not exist """ return True for i in range(R)]) if existence: return True M = hadamard_matrix_paleyI(n) else: raise ValueError("The Hadamard matrix of order %s is not yet implemented." % n)
def hadamard_matrix_www(url_file, comments=False): """ Pulls file from Sloane's database and returns 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] """ n = eval(url_file.split(".")[1]) rws = [] url = "http://neilsloane.com/hadamard/" + url_file f = urlopen(url) s = f.readlines() for i in range(n): r = [] for j in range(n): if s[i][j] == "+": r.append(1) else: r.append(-1) rws.append(r) f.close() if comments: lastline = s[-1] if lastline[0] == "A": print(lastline) return matrix(rws)
_rshcd_cache = {}
def regular_symmetric_hadamard_matrix_with_constant_diagonal(n,e,existence=False): r""" 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 [HX10]_ or 10.5.1 in [BH12]_. For the case `n=324`, see :func:`RSHCD_324` and [CP16]_.
INPUT:
- ``n`` (integer) -- side of the matrix
- ``e`` -- one of `-1` or `+1`, equal to 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:
.. [BH12] \A. Brouwer and W. Haemers, Spectra of graphs, Springer, 2012, http://homepages.cwi.nl/~aeb/math/ipm/ipm.pdf
.. [HX10] \W. Haemers and Q. Xiang, Strongly regular graphs with parameters `(4m^4,2m^4+m^2,m^4+m^2,m^4+m^2)` exist for all `m>1`, European Journal of Combinatorics, Volume 31, Issue 6, August 2010, Pages 1553-1559, :doi:`10.1016/j.ejc.2009.07.009` """
raise ValueError if existence: return False raise ValueError else: else: M = strongly_regular_graph(36,14,4,6).adjacency_matrix() M = -J(36) + 2*M + 2*I(36) if existence: return true() if e == -1: M = strongly_regular_graph(100,44,18,20).adjacency_matrix() M = 2*M - J(100) + 2*I(100) else: M = strongly_regular_graph(100,45,20,20).adjacency_matrix() M = J(100) - 2*M M = strongly_regular_graph(196,91,42,42).adjacency_matrix() M = J(196) - 2*M if existence: return true() M = RSHCD_324(e) n%16 == 0 and not sqn is None and is_prime_power(sqn-1) and is_prime_power(sqn+1)):
not sqn is None and sqn%4 == 2 and True == strongly_regular_graph(sqn-1,(sqn-2)//2,(sqn-6)//4, existence=True) and is_prime_power(ZZ(sqn+1))): if existence: return true() M = rshcd_from_prime_power_and_conference_matrix(sqn+1)
# Recursive construction: the kronecker product of two RSHCD is a RSHCD else: regular_symmetric_hadamard_matrix_with_constant_diagonal(n2,e2,existence=True)):
raise ValueError("I do not know how to build a {}-RSHCD".format((n,e)))
def RSHCD_324(e): r""" 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 :meth:`JankoKharaghaniTonchevGraph <sage.graphs.graph_generators.GraphGenerators.JankoKharaghaniTonchevGraph>` and for the case `\epsilon=-1` from the "twist" `M'` of `M`, using Lemma 11 in [HX10]_. Namely, it turns out that the matrix
.. MATH::
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},
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. [CP16]_.
INPUT:
- ``e`` -- one of `-1` or `+1`, equal to the value of `\epsilon`
TESTS::
sage: from sage.combinat.matrices.hadamard_matrix import RSHCD_324, is_hadamard_matrix sage: for e in [1,-1]: # long time ....: M = RSHCD_324(e) ....: print("{} {} {}".format(M==M.T,is_hadamard_matrix(M),all([M[i,i]==1 for i in range(324)]))) ....: print(set(map(sum,M))) True True True set([18]) True True True set([-18])
REFERENCE:
.. [CP16] \N. Cohen, D. Pasechnik, Implementing Brouwer's database of strongly regular graphs, Designs, Codes, and Cryptography, 2016 :doi:`10.1007/s10623-016-0264-x` """
from sage.graphs.generators.smallgraphs import JankoKharaghaniTonchevGraph as JKTG M = JKTG().adjacency_matrix() M = J(324) - 2*M if e==-1: M1=M[:162].T M2=M[162:].T M11=M1[:162] M12=M1[162:].T M21=M2[:162].T M=block_matrix([[M12,-M11],[-M11.T,M21]]) return M
def _helper_payley_matrix(n, zero_position=True): r""" Return the matrix constructed in Lemma 1.19 page 291 of [SWW72]_.
This function return a `n^2` matrix `M` whose rows/columns are indexed by the element of a finite field on `n` elements `x_1,...,x_n`. The value `M_{i,j}` is equal to `\chi(x_i-x_j)`.
The elements `x_1,...,x_n` are ordered in such a way that the matrix (respectively, its submatrix obtained by removing first row and first column in the case ``zero_position=False``) is symmetric with respect to its second diagonal. The matrix is symmetric if `n=4k+1`, and skew-symmetric otherwise.
INPUT:
- ``n`` -- an odd prime power.
- ``zero_position`` -- if it is true (default), place 0 of ``F_n`` in the middle, otherwise place it first.
.. SEEALSO::
:func:`rshcd_from_close_prime_powers`
EXAMPLES::
sage: from sage.combinat.matrices.hadamard_matrix import _helper_payley_matrix sage: _helper_payley_matrix(5) [ 0 1 -1 -1 1] [ 1 0 1 -1 -1] [-1 1 0 1 -1] [-1 -1 1 0 1] [ 1 -1 -1 1 0]
TESTS::
sage: _helper_payley_matrix(11,zero_position=True) [ 0 -1 1 -1 -1 -1 1 1 1 -1 1] [ 1 0 -1 -1 1 -1 1 -1 1 1 -1] [-1 1 0 1 -1 -1 -1 -1 1 1 1] [ 1 1 -1 0 1 -1 -1 1 -1 -1 1] [ 1 -1 1 -1 0 1 -1 -1 -1 1 1] [ 1 1 1 1 -1 0 1 -1 -1 -1 -1] [-1 -1 1 1 1 -1 0 1 -1 1 -1] [-1 1 1 -1 1 1 -1 0 1 -1 -1] [-1 -1 -1 1 1 1 1 -1 0 -1 1] [ 1 -1 -1 1 -1 1 -1 1 1 0 -1] [-1 1 -1 -1 -1 1 1 1 -1 1 0] sage: _helper_payley_matrix(11,zero_position=False) [ 0 1 1 1 1 -1 1 -1 -1 -1 -1] [-1 0 -1 1 -1 -1 1 1 1 -1 1] [-1 1 0 -1 -1 1 1 -1 1 1 -1] [-1 -1 1 0 1 -1 -1 -1 1 1 1] [-1 1 1 -1 0 1 -1 1 -1 -1 1] [ 1 1 -1 1 -1 0 -1 -1 -1 1 1] [-1 -1 -1 1 1 1 0 1 -1 1 -1] [ 1 -1 1 1 -1 1 -1 0 1 -1 -1] [ 1 -1 -1 -1 1 1 1 -1 0 -1 1] [ 1 1 -1 -1 1 -1 -1 1 1 0 -1] [ 1 -1 1 -1 -1 -1 1 1 -1 1 0] """
# Order the elements of K in K_list # so that K_list[i] = -K_list[n-i-1] else:
for x in K_list] for y in K_list])
def rshcd_from_close_prime_powers(n): r""" 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 [GS70]_.
Note that the authors of [SWW72]_ 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}`
.. SEEALSO::
:func:`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:
.. [SWW72] A Street, W. Wallis, J. Wallis, Combinatorics: Room squares, sum-free sets, Hadamard matrices. Lecture notes in Mathematics 292 (1972). """ raise ValueError("n(={}) must be congruent to 0 mod 4")
def williamson_goethals_seidel_skew_hadamard_matrix(a, b, c, d, check=True): r""" 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 specifying the 1st row of `A`
- ``b`` -- 1,-1 list specifying the 1st row of `B`
- ``d`` -- 1,-1 list specifying the 1st row of `C`
- ``c`` -- 1,-1 list specifying the 1st row of `D`
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] \J.M. Goethals and J. J. Seidel, A skew Hadamard matrix of order 36, J. Aust. Math. Soc. 11(1970), 343-344 .. [Wall71] \J. Wallis, A skew-Hadamard matrix of order 92, Bull. Aust. Math. Soc. 5(1971), 203-204 .. [KoSt08] \C. Koukouvinos, S. Stylianou On skew-Hadamard matrices, Discrete Math. 308(2008) 2723-2731
"""
[-B*R, A, -D.T*R, C.T*R], [-C*R, D.T*R, A, -B.T*R], [-D*R, -C.T*R, B.T*R, A]])
def GS_skew_hadamard_smallcases(n, existence=False, check=True): r""" 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 :func:`sage.combinat.matrices.hadamard_matrix.williamson_goethals_seidel_skew_hadamard_matrix` Matrices for `n=36` and `52` are given in [GS70s]_. Matrices for `n=92` are given in [Wall71]_.
INPUT:
- ``n`` -- the order of the matrix
- ``existence`` -- if true (default), only check that we can do the construction
- ``check`` -- if true (default), check the result.
TESTS::
sage: from sage.combinat.matrices.hadamard_matrix import GS_skew_hadamard_smallcases sage: GS_skew_hadamard_smallcases(36) 36 x 36 dense matrix over Integer Ring... sage: GS_skew_hadamard_smallcases(52) 52 x 52 dense matrix over Integer Ring... sage: GS_skew_hadamard_smallcases(92) 92 x 92 dense matrix over Integer Ring... sage: GS_skew_hadamard_smallcases(100) """ williamson_goethals_seidel_skew_hadamard_matrix as WGS
_skew_had_cache={}
def skew_hadamard_matrix(n,existence=False, skew_normalize=True, check=True): r""" 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 constructions from Section 14.1 of [Ha83]_ and few more exotic ones are implemented.
INPUT:
- ``n`` (integer) -- dimension of the matrix
- ``existence`` (boolean) -- 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 :mod:`sage.misc.unknown`).
- ``False`` -- meaning that the matrix does not exist.
- ``skew_normalize`` (boolean) -- whether to make the 1st row all-one, and adjust the 1st column accordingly. Set to ``True`` by default.
- ``check`` (boolean) -- 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. Set to ``True`` by default.
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]
TESTS::
sage: skew_hadamard_matrix(10,existence=True) False sage: skew_hadamard_matrix(12,existence=True) True sage: skew_hadamard_matrix(784,existence=True) True sage: skew_hadamard_matrix(10) Traceback (most recent call last): ... ValueError: A skew Hadamard matrix of order 10 does not exist sage: skew_hadamard_matrix(36) 36 x 36 dense matrix over Integer Ring... sage: skew_hadamard_matrix(36)==skew_hadamard_matrix(36,skew_normalize=False) False sage: skew_hadamard_matrix(52) 52 x 52 dense matrix over Integer Ring... sage: skew_hadamard_matrix(92) 92 x 92 dense matrix over Integer Ring... sage: skew_hadamard_matrix(816) # long time 816 x 816 dense matrix over Integer Ring... sage: skew_hadamard_matrix(100) Traceback (most recent call last): ... ValueError: A skew Hadamard matrix of order 100 is not yet implemented. sage: skew_hadamard_matrix(100,existence=True) Unknown
REFERENCES:
.. [Ha83] \M. Hall, Combinatorial Theory, 2nd edition, Wiley, 1983 """ return true() return true()
else: # try Williamson construction (Lemma 14.1.5 in [Ha83]_) and skew_hadamard_matrix(n1,existence=True): H = skew_hadamard_matrix(n1, check=False)-I(n1) U = matrix(ZZ, d, lambda i, j: -1 if i==j==0 else\ 1 if i==j==1 or (i>1 and j-1==d-i)\ else 0) A = block_matrix([[matrix([0]), matrix(ZZ,1,d-1,[1]*(d-1))], [ matrix(ZZ,d-1,1,[-1]*(d-1)), _helper_payley_matrix(d-1,zero_position=0)]])+I(d) M = A.tensor_product(I(n1))+(U*A).tensor_product(H) break return true()
else:
def symmetric_conference_matrix(n, check=True): r""" 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 [BH12]_. 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) -- 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. Set to ``True`` by default.
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 """ except ValueError: raise
def szekeres_difference_set_pair(m, check=True): r""" Construct Szekeres `(2m+1,m,1)`-cyclic difference family
Let `4m+3` be a prime power. Theorem 3 in [Sz69]_ 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 [SWW72]_ (there the formula for `B` is correct, as opposed to (4.2) in [Sz69]_, where the sign before `1` is wrong.
In modern terminology, for `m>1` the sets `A` and `B` form a :func:`difference family<sage.combinat.designs.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`` (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:
.. [Sz69] \G. Szekeres, Tournaments and Hadamard matrices, Enseignement Math. (2) 15(1969), 269-278 """
def typeI_matrix_difference_set(G,A): r""" (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] """
def rshcd_from_prime_power_and_conference_matrix(n): r""" 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 [WW72]_.
In [SWW72]_ this construction (Theorem 5.15 and Corollary 5.16) is reproduced with a typo. Note that [WW72]_ refers to [Sz69]_ for the construction, provided by :func:`szekeres_difference_set_pair`, of complementary difference sets, and the latter has a typo.
From a :func:`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
.. SEEALSO::
:func:`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:
.. [WW72] \J. Wallis and A.L. Whiteman, Some classes of Hadamard matrices with constant diagonal, Bull. Austral. Math. Soc. 7(1972), 233-249 """ except ValueError: return [J(1,1), f, e_t_f, -e_t_f], [f.T, J4m, e.tensor_product(W-II), e.tensor_product(W+II)], [ e_t_f.T, (e.T).tensor_product(W-II), A_t_W+JJ.tensor_product(II), H34], [-e_t_f.T, (e.T).tensor_product(W+II), H34.T, -A_t_W+JJ.tensor_product(II)]]) |