def _helper_payley_matrix(n):
r"""
Return the marix 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)`. Note that `n` must be an odd prime power.
The elements `x_1,...,x_n` are ordered in such a way that the matrix is
symmetric with respect to its second diagonal. The matrix is symmetric if
n==4k+1, and skew-symmetric if n=4k-1.
INPUT:
- ``n`` -- a prime power
.. SEEALSO::
:func:`rshcd_from_close_prime_powers`
EXAMPLE::
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]
"""
from sage.rings.finite_rings.constructor import FiniteField as GF
K = GF(n,conway=True,prefix='x')
# Order the elements of K in K_list
# so that K_list[i] = -K_list[n-i-1]
K_pairs = set(frozenset([x,-x]) for x in K)
K_pairs.discard(frozenset([0]))
K_list = [None]*n
for i,(x,y) in enumerate(K_pairs):
K_list[i] = x
K_list[-i-1] = y
K_list[n//2] = K(0)
M = matrix(n,[[2*((x-y).is_square())-1
for x in K_list]
for y in K_list])
M = M-I(n)
assert (M*J(n)).is_zero()
assert (M*M.transpose()) == n*I(n)-J(n)
return M
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 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).
"""
if n % 4:
raise ValueError("n(={}) must be congruent to 0 mod 4")
a, b = sorted([n - 1, n + 1], key=lambda x: -x % 4)
Sa = _helper_payley_matrix(a)
Sb = _helper_payley_matrix(b)
U = matrix(a, [[int(i + j == a - 1) for i in range(a)] for j in range(a)])
K = (U * Sa).tensor_product(Sb) + U.tensor_product(J(b) - I(b)) - J(
a).tensor_product(I(b))
F = lambda x: diagonal_matrix([-(-1)**i for i in range(x)])
G = block_diagonal_matrix([J(1), I(a).tensor_product(F(b))])
e = matrix(a * b, [1] * (a * b))
H = block_matrix(2, [-J(1), e.transpose(), e, K])
HH = G * H * G
assert len(set(map(sum, HH))) == 1
assert HH**2 == n**2 * I(n**2)
return HH
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]
"""
from sage.rings.finite_rings.finite_field_constructor import FiniteField as GF
K = GF(n, prefix='x')
# Order the elements of K in K_list
# so that K_list[i] = -K_list[n-i-1]
K_pairs = set(frozenset([x, -x]) for x in K)
K_pairs.discard(frozenset([0]))
K_list = [None] * n
if zero_position:
zero_position = n // 2
shift = 0
else:
shift = 1
for i, (x, y) in enumerate(K_pairs):
K_list[i + shift] = x
K_list[-i - 1] = y
K_list[zero_position] = K(0)
M = matrix(n, [[2 * ((x - y).is_square()) - 1 for x in K_list]
for y in K_list])
M = M - I(n)
assert (M * J(n)).is_zero()
assert (M * M.transpose()) == n * I(n) - J(n)
return M
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`
"""
if existence and (n, e) in _rshcd_cache:
return _rshcd_cache[n, e]
from sage.graphs.strongly_regular_db import strongly_regular_graph
def true():
_rshcd_cache[n, e] = True
return True
M = None
if abs(e) != 1:
raise ValueError
sqn = None
if is_square(n):
sqn = int(sqrt(n))
if n < 0:
if existence:
return False
raise ValueError
elif n == 4:
if existence:
return true()
if e == 1:
M = J(4) - 2 * matrix(4, [[int(i + j == 3) for i in range(4)]
for j in range(4)])
else:
M = -J(4) + 2 * I(4)
elif n == 36:
if existence:
return true()
if e == 1:
M = strongly_regular_graph(36, 15, 6, 6).adjacency_matrix()
M = J(36) - 2 * M
else:
M = strongly_regular_graph(36, 14, 4, 6).adjacency_matrix()
M = -J(36) + 2 * M + 2 * I(36)
elif n == 100:
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
elif n == 196 and e == 1:
if existence:
return true()
M = strongly_regular_graph(196, 91, 42, 42).adjacency_matrix()
M = J(196) - 2 * M
elif n == 324:
if existence:
return true()
M = RSHCD_324(e)
elif (e == 1 and n % 16 == 0 and not sqn is None
and is_prime_power(sqn - 1) and is_prime_power(sqn + 1)):
if existence:
return true()
M = -rshcd_from_close_prime_powers(sqn)
elif (e == 1 and 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:
from itertools import product
for n1, e1 in product(divisors(n)[1:-1], [-1, 1]):
e2 = e1 * e
n2 = n // n1
if (regular_symmetric_hadamard_matrix_with_constant_diagonal(
n1, e1, existence=True) and
regular_symmetric_hadamard_matrix_with_constant_diagonal(
n2, e2, existence=True)):
if existence:
return true()
M1 = regular_symmetric_hadamard_matrix_with_constant_diagonal(
n1, e1)
M2 = regular_symmetric_hadamard_matrix_with_constant_diagonal(
n2, e2)
M = M1.tensor_product(M2)
break
if M is None:
from sage.misc.unknown import Unknown
_rshcd_cache[n, e] = Unknown
if existence:
return Unknown
raise ValueError("I do not know how to build a {}-RSHCD".format(
(n, e)))
assert M * M.transpose() == n * I(n)
assert set(map(sum, M)) == {ZZ(e * sqn)}
return M
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
"""
from sage.graphs.strongly_regular_db import strongly_regular_graph as srg
if is_prime_power(n) and 2 == (n - 1) % 4:
try:
M = srg(n - 2, (n - 3) // 2, (n - 7) // 4)
except ValueError:
return
m = (n - 3) // 4
Q, X, Y = szekeres_difference_set_pair(m)
B = typeI_matrix_difference_set(Q, X)
A = -typeI_matrix_difference_set(Q, Y) # must be symmetric
W = M.seidel_adjacency_matrix()
f = J(1, 4 * m + 1)
e = J(1, 2 * m + 1)
JJ = J(2 * m + 1, 2 * m + 1)
II = I(n - 2)
Ib = I(2 * m + 1)
J4m = J(4 * m + 1, 4 * m + 1)
H34 = -(B + Ib).tensor_product(W) + Ib.tensor_product(J4m) + (
Ib - JJ).tensor_product(II)
A_t_W = A.tensor_product(W)
e_t_f = e.tensor_product(f)
H = block_matrix(
[[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)
]])
return H
