def ProjectivePlaneDesign(n, type="Desarguesian", algorithm=None):
r"""
Returns a projective plane of order `n`.
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`.
INPUT:
- ``n`` -- the finite projective plane's order
- ``type`` -- When set to ``"Desarguesian"``, the method returns
Desarguesian projective planes, i.e. a finite projective plane obtained by
considering the 1- and 2- dimensional spaces of `F_n^3`.
For the moment, no other value is available for this parameter.
- ``algorithm`` -- set to ``None`` by default, which results in using Sage's
own implementation. In order to use GAP's implementation instead (i.e. its
``PGPointFlatBlockDesign`` function) set ``algorithm="gap"``. Note that
GAP's "design" package must be available in this case, and that it can be
installed with the ``gap_packages`` spkg.
.. SEEALSO::
:meth:`ProjectiveGeometryDesign`
EXAMPLES::
sage: designs.ProjectivePlaneDesign(2)
Incidence structure with 7 points and 7 blocks
Non-existent ones::
sage: designs.ProjectivePlaneDesign(10)
Traceback (most recent call last):
...
ValueError: No projective plane design of order 10 exists.
sage: designs.ProjectivePlaneDesign(14)
Traceback (most recent call last):
...
ValueError: By the Bruck-Ryser-Chowla theorem, no projective plane of order 14 exists.
An unknown one::
sage: designs.ProjectivePlaneDesign(12)
Traceback (most recent call last):
...
ValueError: If such a projective plane exists, we do not know how to build it.
TESTS::
sage: designs.ProjectivePlaneDesign(10, type="AnyThingElse")
Traceback (most recent call last):
...
ValueError: The value of 'type' must be 'Desarguesian'.
sage: designs.ProjectivePlaneDesign(2, algorithm="gap") # optional - gap_packages
Incidence structure with 7 points and 7 blocks
"""
from sage.rings.arith import two_squares
if type != "Desarguesian":
raise ValueError("The value of 'type' must be 'Desarguesian'.")
try:
F = FiniteField(n, 'x')
except ValueError:
if n == 10:
raise ValueError("No projective plane design of order 10 exists.")
try:
if (n%4) in [1,2]:
two_squares(n)
except ValueError:
raise ValueError("By the Bruck-Ryser-Chowla theorem, no projective"
" plane of order "+str(n)+" exists.")
raise ValueError("If such a projective plane exists, "
"we do not know how to build it.")
return ProjectiveGeometryDesign(2,1,F, algorithm=algorithm)
def projective_plane(n, check=True, existence=False):
r"""
Returns 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)
Incidence structure with 7 points and 7 blocks
sage: designs.projective_plane(3)
Incidence structure with 13 points and 13 blocks
sage: designs.projective_plane(4)
Incidence structure with 21 points and 21 blocks
sage: designs.projective_plane(5)
Incidence structure with 31 points and 31 blocks
sage: designs.projective_plane(6)
Traceback (most recent call last):
...
EmptySetError: By the Ryser-Chowla 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 Ryser-Chowla 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.arith import is_prime_power, two_squares
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]:
try:
two_squares(n)
except ValueError:
if existence:
return False
raise EmptySetError("By the Ryser-Chowla theorem, no projective"
" plane of order "+str(n)+" exists.")
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, check=check)
def ProjectivePlaneDesign(n, type="Desarguesian"):
r"""
Returns a projective plane of order `n`.
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`.
INPUT:
- ``n`` -- the finite projective plane's order
- ``type`` -- When set to ``"Desarguesian"``, the method returns
Desarguesian projective planes, i.e. a finite projective plane obtained by
considering the 1- and 2- dimensional spaces of `F_n^3`.
For the moment, no other value is available for this parameter.
.. SEEALSO::
:meth:`ProjectiveGeometryDesign`
EXAMPLES::
sage: designs.ProjectivePlaneDesign(2)
Incidence structure with 7 points and 7 blocks
Non-existent ones::
sage: designs.ProjectivePlaneDesign(10)
Traceback (most recent call last):
...
ValueError: No projective plane design of order 10 exists.
sage: designs.ProjectivePlaneDesign(14)
Traceback (most recent call last):
...
ValueError: By the Bruck-Ryser-Chowla theorem, no projective plane of order 14 exists.
An unknown one::
sage: designs.ProjectivePlaneDesign(12)
Traceback (most recent call last):
...
ValueError: If such a projective plane exists, we do not know how to build it.
TESTS::
sage: designs.ProjectivePlaneDesign(10, type="AnyThingElse")
Traceback (most recent call last):
...
ValueError: The value of 'type' must be 'Desarguesian'.
"""
from sage.rings.arith import two_squares
if type != "Desarguesian":
raise ValueError("The value of 'type' must be 'Desarguesian'.")
try:
F = FiniteField(n, 'x')
except ValueError:
if n == 10:
raise ValueError("No projective plane design of order 10 exists.")
try:
if (n%4) in [1,2]:
two_squares(n)
except ValueError:
raise ValueError("By the Bruck-Ryser-Chowla theorem, no projective"
" plane of order "+str(n)+" exists.")
raise ValueError("If such a projective plane exists, "
"we do not know how to build it.")
return ProjectiveGeometryDesign(2,1,F)
def ProjectivePlaneDesign(n, type="Desarguesian", algorithm=None):
r"""
Returns a projective plane of order `n`.
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`.
INPUT:
- ``n`` -- the finite projective plane's order
- ``type`` -- When set to ``"Desarguesian"``, the method returns
Desarguesian projective planes, i.e. a finite projective plane obtained by
considering the 1- and 2- dimensional spaces of `F_n^3`.
For the moment, no other value is available for this parameter.
- ``algorithm`` -- set to ``None`` by default, which results in using Sage's
own implementation. In order to use GAP's implementation instead (i.e. its
``PGPointFlatBlockDesign`` function) set ``algorithm="gap"``. Note that
GAP's "design" package must be available in this case, and that it can be
installed with the ``gap_packages`` spkg.
.. SEEALSO::
:meth:`ProjectiveGeometryDesign`
EXAMPLES::
sage: designs.ProjectivePlaneDesign(2)
Incidence structure with 7 points and 7 blocks
Non-existent ones::
sage: designs.ProjectivePlaneDesign(10)
Traceback (most recent call last):
...
ValueError: No projective plane design of order 10 exists.
sage: designs.ProjectivePlaneDesign(14)
Traceback (most recent call last):
...
ValueError: By the Bruck-Ryser-Chowla theorem, no projective plane of order 14 exists.
An unknown one::
sage: designs.ProjectivePlaneDesign(12)
Traceback (most recent call last):
...
ValueError: If such a projective plane exists, we do not know how to build it.
TESTS::
sage: designs.ProjectivePlaneDesign(10, type="AnyThingElse")
Traceback (most recent call last):
...
ValueError: The value of 'type' must be 'Desarguesian'.
sage: designs.ProjectivePlaneDesign(2, algorithm="gap") # optional - gap_packages
Incidence structure with 7 points and 7 blocks
"""
from sage.rings.arith import two_squares
if type != "Desarguesian":
raise ValueError("The value of 'type' must be 'Desarguesian'.")
try:
F = FiniteField(n, 'x')
except ValueError:
if n == 10:
raise ValueError("No projective plane design of order 10 exists.")
try:
if (n % 4) in [1, 2]:
two_squares(n)
except ValueError:
raise ValueError("By the Bruck-Ryser-Chowla theorem, no projective"
" plane of order " + str(n) + " exists.")
raise ValueError("If such a projective plane exists, "
"we do not know how to build it.")
return ProjectiveGeometryDesign(2, 1, F, algorithm=algorithm)
def projective_plane(n, check=True, existence=False):
r"""
Returns 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)
Incidence structure with 7 points and 7 blocks
sage: designs.projective_plane(3)
Incidence structure with 13 points and 13 blocks
sage: designs.projective_plane(4)
Incidence structure with 21 points and 21 blocks
sage: designs.projective_plane(5)
Incidence structure with 31 points and 31 blocks
sage: designs.projective_plane(6)
Traceback (most recent call last):
...
EmptySetError: By the Ryser-Chowla 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 Ryser-Chowla 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.arith import is_prime_power, two_squares
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]:
try:
two_squares(n)
except ValueError:
if existence:
return False
raise EmptySetError("By the Ryser-Chowla theorem, no projective"
" plane of order " + str(n) + " exists.")
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, check=check)
