def projective_plane(n, check=True, existence=False):
r"""
Return a projective plane of order ``n`` as a 2-design.
A finite projective plane is a 2-design with `n^2+n+1` lines (or blocks) and
`n^2+n+1` points. For more information on finite projective planes, see the
:wikipedia:`Projective_plane#Finite_projective_planes`.
If no construction is possible, then the function raises a ``EmptySetError``
whereas if no construction is available the function raises a
``NotImplementedError``.
INPUT:
- ``n`` -- the finite projective plane's order
EXAMPLES::
sage: designs.projective_plane(2)
(7,3,1)-Balanced Incomplete Block Design
sage: designs.projective_plane(3)
(13,4,1)-Balanced Incomplete Block Design
sage: designs.projective_plane(4)
(21,5,1)-Balanced Incomplete Block Design
sage: designs.projective_plane(5)
(31,6,1)-Balanced Incomplete Block Design
sage: designs.projective_plane(6)
Traceback (most recent call last):
...
EmptySetError: By the Bruck-Ryser theorem, no projective plane of order 6 exists.
sage: designs.projective_plane(10)
Traceback (most recent call last):
...
EmptySetError: No projective plane of order 10 exists by C. Lam, L. Thiel and S. Swiercz "The nonexistence of finite projective planes of order 10" (1989), Canad. J. Math.
sage: designs.projective_plane(12)
Traceback (most recent call last):
...
NotImplementedError: If such a projective plane exists, we do not know how to build it.
sage: designs.projective_plane(14)
Traceback (most recent call last):
...
EmptySetError: By the Bruck-Ryser theorem, no projective plane of order 14 exists.
TESTS::
sage: designs.projective_plane(2197, existence=True)
True
sage: designs.projective_plane(6, existence=True)
False
sage: designs.projective_plane(10, existence=True)
False
sage: designs.projective_plane(12, existence=True)
Unknown
"""
from sage.rings.sum_of_squares import is_sum_of_two_squares_pyx
if n <= 1:
if existence:
return False
raise EmptySetError("There is no projective plane of order <= 1")
if n == 10:
if existence:
return False
ref = ("C. Lam, L. Thiel and S. Swiercz \"The nonexistence of finite "
"projective planes of order 10\" (1989), Canad. J. Math.")
raise EmptySetError("No projective plane of order 10 exists by %s"%ref)
if (n%4) in [1,2] and not is_sum_of_two_squares_pyx(n):
if existence:
return False
raise EmptySetError("By the Bruck-Ryser theorem, no projective"
" plane of order {} exists.".format(n))
if not is_prime_power(n):
if existence:
return Unknown
raise NotImplementedError("If such a projective plane exists, we do "
"not know how to build it.")
if existence:
return True
else:
return DesarguesianProjectivePlaneDesign(n, point_coordinates=False, check=check)
def projective_plane(n, check=True, existence=False):
r"""
Return a projective plane of order ``n`` as a 2-design.
A finite projective plane is a 2-design with `n^2+n+1` lines (or blocks) and
`n^2+n+1` points. For more information on finite projective planes, see the
:wikipedia:`Projective_plane#Finite_projective_planes`.
If no construction is possible, then the function raises a ``EmptySetError``
whereas if no construction is available the function raises a
``NotImplementedError``.
INPUT:
- ``n`` -- the finite projective plane's order
EXAMPLES::
sage: designs.projective_plane(2)
(7,3,1)-Balanced Incomplete Block Design
sage: designs.projective_plane(3)
(13,4,1)-Balanced Incomplete Block Design
sage: designs.projective_plane(4)
(21,5,1)-Balanced Incomplete Block Design
sage: designs.projective_plane(5)
(31,6,1)-Balanced Incomplete Block Design
sage: designs.projective_plane(6)
Traceback (most recent call last):
...
EmptySetError: By the Bruck-Ryser theorem, no projective plane of order 6 exists.
sage: designs.projective_plane(10)
Traceback (most recent call last):
...
EmptySetError: No projective plane of order 10 exists by C. Lam, L. Thiel and S. Swiercz "The nonexistence of finite projective planes of order 10" (1989), Canad. J. Math.
sage: designs.projective_plane(12)
Traceback (most recent call last):
...
NotImplementedError: If such a projective plane exists, we do not know how to build it.
sage: designs.projective_plane(14)
Traceback (most recent call last):
...
EmptySetError: By the Bruck-Ryser theorem, no projective plane of order 14 exists.
TESTS::
sage: designs.projective_plane(2197, existence=True)
True
sage: designs.projective_plane(6, existence=True)
False
sage: designs.projective_plane(10, existence=True)
False
sage: designs.projective_plane(12, existence=True)
Unknown
"""
from sage.rings.sum_of_squares import is_sum_of_two_squares_pyx
if n <= 1:
if existence:
return False
raise EmptySetError("There is no projective plane of order <= 1")
if n == 10:
if existence:
return False
ref = ("C. Lam, L. Thiel and S. Swiercz \"The nonexistence of finite "
"projective planes of order 10\" (1989), Canad. J. Math.")
raise EmptySetError("No projective plane of order 10 exists by %s"%ref)
if (n%4) in [1,2] and not is_sum_of_two_squares_pyx(n):
if existence:
return False
raise EmptySetError("By the Bruck-Ryser theorem, no projective"
" plane of order {} exists.".format(n))
if not is_prime_power(n):
if existence:
return Unknown
raise NotImplementedError("If such a projective plane exists, we do "
"not know how to build it.")
if existence:
return True
else:
return DesarguesianProjectivePlaneDesign(n, point_coordinates=False, check=check)
