sage.combinat.designs: Update from #35095

Author: Matthias Koeppe

src/sage/combinat/designs/block_design.py

@@ -54,14 +54,17 @@
 #*****************************************************************************
 from sage.arith.misc import binomial, integer_floor, is_prime_power
 from sage.categories.sets_cat import EmptySetError
-from sage.modules.free_module import VectorSpace
+from sage.misc.lazy_import import lazy_import
+from sage.misc.unknown import Unknown
 from sage.rings.integer import Integer
 from sage.rings.integer_ring import ZZ
+
 from .incidence_structures import IncidenceStructure
-from sage.rings.finite_rings.finite_field_constructor import FiniteField
-from sage.misc.unknown import Unknown
-from sage.matrix.matrix_space import MatrixSpace
-from sage.libs.gap.libgap import libgap
+
+lazy_import('sage.libs.gap.libgap', 'libgap')
+lazy_import('sage.matrix.matrix_space', 'MatrixSpace')
+lazy_import('sage.modules.free_module', 'VectorSpace')
+lazy_import('sage.rings.finite_rings.finite_field_constructor', 'FiniteField')
 
 
 BlockDesign = IncidenceStructure
@@ -214,30 +217,30 @@ def ProjectiveGeometryDesign(n, d, F, algorithm=None, point_coordinates=True, ch
     The set of `d`-dimensional subspaces in a `n`-dimensional projective space
     forms `2`-designs (or balanced incomplete block designs)::
 
-        sage: PG = designs.ProjectiveGeometryDesign(4, 2, GF(2)); PG                    # optional - sage.rings.finite_rings
+        sage: PG = designs.ProjectiveGeometryDesign(4, 2, GF(2)); PG
         Incidence structure with 31 points and 155 blocks
-        sage: PG.is_t_design(return_parameters=True)                                    # optional - sage.rings.finite_rings
+        sage: PG.is_t_design(return_parameters=True)
         (True, (2, 31, 7, 7))
 
-        sage: PG = designs.ProjectiveGeometryDesign(3, 1, GF(4))                        # optional - sage.rings.finite_rings
-        sage: PG.is_t_design(return_parameters=True)                                    # optional - sage.rings.finite_rings
+        sage: PG = designs.ProjectiveGeometryDesign(3, 1, GF(4))
+        sage: PG.is_t_design(return_parameters=True)
         (True, (2, 85, 5, 1))
 
     Check with ``F`` being a prime power::
 
-        sage: PG = designs.ProjectiveGeometryDesign(3, 2, 4); PG                        # optional - sage.rings.finite_rings
+        sage: PG = designs.ProjectiveGeometryDesign(3, 2, 4); PG
         Incidence structure with 85 points and 85 blocks
 
     Use coordinates::
 
-        sage: PG = designs.ProjectiveGeometryDesign(2, 1, GF(3))                        # optional - sage.rings.finite_rings
-        sage: PG.blocks()[0]                                                            # optional - sage.rings.finite_rings
+        sage: PG = designs.ProjectiveGeometryDesign(2, 1, GF(3))
+        sage: PG.blocks()[0]
         [(1, 0, 0), (1, 0, 1), (1, 0, 2), (0, 0, 1)]
 
     Use indexing by integers::
 
-        sage: PG = designs.ProjectiveGeometryDesign(2, 1, GF(3), point_coordinates=0)   # optional - sage.rings.finite_rings
-        sage: PG.blocks()[0]                                                            # optional - sage.rings.finite_rings
+        sage: PG = designs.ProjectiveGeometryDesign(2, 1, GF(3), point_coordinates=0)
+        sage: PG.blocks()[0]
         [0, 1, 2, 12]
 
     Check that the constructor using gap also works::
@@ -318,15 +321,15 @@ def DesarguesianProjectivePlaneDesign(n, point_coordinates=True, check=True):
 
     EXAMPLES::
 
-        sage: designs.DesarguesianProjectivePlaneDesign(2)                              # optional - sage.rings.finite_rings
+        sage: designs.DesarguesianProjectivePlaneDesign(2)
         (7,3,1)-Balanced Incomplete Block Design
-        sage: designs.DesarguesianProjectivePlaneDesign(3)                              # optional - sage.rings.finite_rings
+        sage: designs.DesarguesianProjectivePlaneDesign(3)
         (13,4,1)-Balanced Incomplete Block Design
-        sage: designs.DesarguesianProjectivePlaneDesign(4)                              # optional - sage.rings.finite_rings
+        sage: designs.DesarguesianProjectivePlaneDesign(4)
         (21,5,1)-Balanced Incomplete Block Design
-        sage: designs.DesarguesianProjectivePlaneDesign(5)                              # optional - sage.rings.finite_rings
+        sage: designs.DesarguesianProjectivePlaneDesign(5)
         (31,6,1)-Balanced Incomplete Block Design
-        sage: designs.DesarguesianProjectivePlaneDesign(6)                              # optional - sage.rings.finite_rings
+        sage: designs.DesarguesianProjectivePlaneDesign(6)
         Traceback (most recent call last):
         ...
         ValueError: the order of a finite field must be a prime power
@@ -400,20 +403,20 @@ def q3_minus_one_matrix(K):
     EXAMPLES::
 
         sage: from sage.combinat.designs.block_design import q3_minus_one_matrix
-        sage: m = q3_minus_one_matrix(GF(3))                                            # optional - sage.rings.finite_rings
-        sage: m.multiplicative_order() == 3**3 - 1                                      # optional - sage.rings.finite_rings
+        sage: m = q3_minus_one_matrix(GF(3))
+        sage: m.multiplicative_order() == 3**3 - 1
         True
 
-        sage: m = q3_minus_one_matrix(GF(4, 'a'))                                       # optional - sage.rings.finite_rings
-        sage: m.multiplicative_order() == 4**3 - 1                                      # optional - sage.rings.finite_rings
+        sage: m = q3_minus_one_matrix(GF(4, 'a'))
+        sage: m.multiplicative_order() == 4**3 - 1
         True
 
-        sage: m = q3_minus_one_matrix(GF(5))                                            # optional - sage.rings.finite_rings
-        sage: m.multiplicative_order() == 5**3 - 1                                      # optional - sage.rings.finite_rings
+        sage: m = q3_minus_one_matrix(GF(5))
+        sage: m.multiplicative_order() == 5**3 - 1
         True
 
-        sage: m = q3_minus_one_matrix(GF(9, 'a'))                                       # optional - sage.rings.finite_rings
-        sage: m.multiplicative_order() == 9**3 - 1                                      # optional - sage.rings.finite_rings
+        sage: m = q3_minus_one_matrix(GF(9, 'a'))
+        sage: m.multiplicative_order() == 9**3 - 1
         True
     """
     q = K.cardinality()
@@ -450,24 +453,24 @@ def normalize_hughes_plane_point(p, q):
 
     INPUT:
 
-    - ``p`` -- point with the coordinates (x,y,z) (a list, a vector, a tuple...)
+    - ``p`` -- point with the coordinates `(x,y,z)` (a list, a vector, a tuple...)
 
     - ``q`` -- cardinality of the underlying finite field
 
     EXAMPLES::
 
         sage: from sage.combinat.designs.block_design import normalize_hughes_plane_point
-        sage: K = FiniteField(9,'x')                                                    # optional - sage.rings.finite_rings
-        sage: x = K.gen()                                                               # optional - sage.rings.finite_rings
-        sage: normalize_hughes_plane_point((x, x+1, x), 9)                              # optional - sage.rings.finite_rings
+        sage: K = FiniteField(9,'x')
+        sage: x = K.gen()
+        sage: normalize_hughes_plane_point((x, x + 1, x), 9)
         (1, x, 1)
-        sage: normalize_hughes_plane_point(vector((x,x,x)), 9)                          # optional - sage.rings.finite_rings
+        sage: normalize_hughes_plane_point(vector((x,x,x)), 9)
         (1, 1, 1)
-        sage: zero = K.zero()                                                           # optional - sage.rings.finite_rings
-        sage: normalize_hughes_plane_point((2*x+2, zero, zero), 9)                      # optional - sage.rings.finite_rings
+        sage: zero = K.zero()
+        sage: normalize_hughes_plane_point((2*x + 2, zero, zero), 9)
         (1, 0, 0)
-        sage: one = K.one()                                                             # optional - sage.rings.finite_rings
-        sage: normalize_hughes_plane_point((2*x, one, zero), 9)                         # optional - sage.rings.finite_rings
+        sage: one = K.one()
+        sage: normalize_hughes_plane_point((2*x, one, zero), 9)
         (2*x, 1, 0)
     """
     for i in [2,1,0]:
@@ -526,7 +529,7 @@ def HughesPlane(q2, check=True):
 
     EXAMPLES::
 
-        sage: H = designs.HughesPlane(9); H                                             # optional - sage.rings.finite_rings
+        sage: H = designs.HughesPlane(9); H
         (91,10,1)-Balanced Incomplete Block Design
 
     We prove in the following computations that the Desarguesian plane ``H`` is
@@ -535,45 +538,45 @@ def HughesPlane(q2, check=True):
     `D_{1,10} \cap D_{70,59}` and `D_{10,0} \cap D_{59,57}` are on the same line
     while `D_{0,70}`, `D_{1,59}` and `D_{10,57}` are not concurrent::
 
-        sage: blocks = H.blocks()                                                       # optional - sage.rings.finite_rings
-        sage: line = lambda p,q: next(b for b in blocks if p in b and q in b)           # optional - sage.rings.finite_rings
+        sage: blocks = H.blocks()
+        sage: line = lambda p,q: next(b for b in blocks if p in b and q in b)
 
-        sage: b_0_1 = line(0, 1)                                                        # optional - sage.rings.finite_rings
-        sage: b_1_10 = line(1, 10)                                                      # optional - sage.rings.finite_rings
-        sage: b_10_0 = line(10, 0)                                                      # optional - sage.rings.finite_rings
-        sage: b_57_70 = line(57, 70)                                                    # optional - sage.rings.finite_rings
-        sage: b_70_59 = line(70, 59)                                                    # optional - sage.rings.finite_rings
-        sage: b_59_57 = line(59, 57)                                                    # optional - sage.rings.finite_rings
+        sage: b_0_1 = line(0, 1)
+        sage: b_1_10 = line(1, 10)
+        sage: b_10_0 = line(10, 0)
+        sage: b_57_70 = line(57, 70)
+        sage: b_70_59 = line(70, 59)
+        sage: b_59_57 = line(59, 57)
 
-        sage: set(b_0_1).intersection(b_57_70)                                          # optional - sage.rings.finite_rings
+        sage: set(b_0_1).intersection(b_57_70)
         {2}
-        sage: set(b_1_10).intersection(b_70_59)                                         # optional - sage.rings.finite_rings
+        sage: set(b_1_10).intersection(b_70_59)
         {73}
-        sage: set(b_10_0).intersection(b_59_57)                                         # optional - sage.rings.finite_rings
+        sage: set(b_10_0).intersection(b_59_57)
         {60}
 
-        sage: line(2, 73) == line(73, 60)                                               # optional - sage.rings.finite_rings
+        sage: line(2, 73) == line(73, 60)
         True
 
-        sage: b_0_57 = line(0, 57)                                                      # optional - sage.rings.finite_rings
-        sage: b_1_70 = line(1, 70)                                                      # optional - sage.rings.finite_rings
-        sage: b_10_59 = line(10, 59)                                                    # optional - sage.rings.finite_rings
+        sage: b_0_57 = line(0, 57)
+        sage: b_1_70 = line(1, 70)
+        sage: b_10_59 = line(10, 59)
 
-        sage: p = set(b_0_57).intersection(b_1_70)                                      # optional - sage.rings.finite_rings
-        sage: q = set(b_1_70).intersection(b_10_59)                                     # optional - sage.rings.finite_rings
-        sage: p == q                                                                    # optional - sage.rings.finite_rings
+        sage: p = set(b_0_57).intersection(b_1_70)
+        sage: q = set(b_1_70).intersection(b_10_59)
+        sage: p == q
         False
 
     TESTS:
 
     Some wrong input::
 
-        sage: designs.HughesPlane(5)                                                    # optional - sage.rings.finite_rings
+        sage: designs.HughesPlane(5)
         Traceback (most recent call last):
         ...
         EmptySetError: No Hughes plane of non-square order exists.
 
-        sage: designs.HughesPlane(16)                                                   # optional - sage.rings.finite_rings
+        sage: designs.HughesPlane(16)
         Traceback (most recent call last):
         ...
         EmptySetError: No Hughes plane of even order exists.
@@ -644,13 +647,11 @@ def projective_plane_to_OA(pplane, pt=None, check=True):
     EXAMPLES::
 
         sage: from sage.combinat.designs.block_design import projective_plane_to_OA
-        sage: p2 = designs.DesarguesianProjectivePlaneDesign(2,                         # optional - sage.rings.finite_rings
-        ....:                                                point_coordinates=False)
-        sage: projective_plane_to_OA(p2)                                                # optional - sage.rings.finite_rings
+        sage: p2 = designs.DesarguesianProjectivePlaneDesign(2, point_coordinates=False)
+        sage: projective_plane_to_OA(p2)
         [[0, 0, 0], [0, 1, 1], [1, 0, 1], [1, 1, 0]]
-        sage: p3 = designs.DesarguesianProjectivePlaneDesign(3,                         # optional - sage.rings.finite_rings
-        ....:                                                point_coordinates=False)
-        sage: projective_plane_to_OA(p3)                                                # optional - sage.rings.finite_rings
+        sage: p3 = designs.DesarguesianProjectivePlaneDesign(3, point_coordinates=False)
+        sage: projective_plane_to_OA(p3)
         [[0, 0, 0, 0],
          [0, 1, 2, 1],
          [0, 2, 1, 2],
@@ -661,11 +662,10 @@ def projective_plane_to_OA(pplane, pt=None, check=True):
          [2, 1, 0, 2],
          [2, 2, 2, 0]]
 
-        sage: pp = designs.DesarguesianProjectivePlaneDesign(16,                        # optional - sage.rings.finite_rings
-        ....:                                                point_coordinates=False)
-        sage: _ = projective_plane_to_OA(pp, pt=0)                                      # optional - sage.rings.finite_rings
-        sage: _ = projective_plane_to_OA(pp, pt=3)                                      # optional - sage.rings.finite_rings
-        sage: _ = projective_plane_to_OA(pp, pt=7)                                      # optional - sage.rings.finite_rings
+        sage: pp = designs.DesarguesianProjectivePlaneDesign(16, point_coordinates=False)
+        sage: _ = projective_plane_to_OA(pp, pt=0)
+        sage: _ = projective_plane_to_OA(pp, pt=3)
+        sage: _ = projective_plane_to_OA(pp, pt=7)
     """
     from .bibd import _relabel_bibd
     pplane = pplane.blocks()
@@ -816,28 +816,28 @@ def AffineGeometryDesign(n, d, F, point_coordinates=True, check=True):
 
     EXAMPLES::
 
-        sage: BD = designs.AffineGeometryDesign(3, 1, GF(2))                            # optional - sage.rings.finite_rings
-        sage: BD.is_t_design(return_parameters=True)                                    # optional - sage.rings.finite_rings
+        sage: BD = designs.AffineGeometryDesign(3, 1, GF(2))
+        sage: BD.is_t_design(return_parameters=True)
         (True, (2, 8, 2, 1))
-        sage: BD = designs.AffineGeometryDesign(3, 2, GF(4))                            # optional - sage.rings.finite_rings
-        sage: BD.is_t_design(return_parameters=True)                                    # optional - sage.rings.finite_rings
+        sage: BD = designs.AffineGeometryDesign(3, 2, GF(4))
+        sage: BD.is_t_design(return_parameters=True)
         (True, (2, 64, 16, 5))
-        sage: BD = designs.AffineGeometryDesign(4, 2, GF(3))                            # optional - sage.rings.finite_rings
-        sage: BD.is_t_design(return_parameters=True)                                    # optional - sage.rings.finite_rings
+        sage: BD = designs.AffineGeometryDesign(4, 2, GF(3))
+        sage: BD.is_t_design(return_parameters=True)
         (True, (2, 81, 9, 13))
 
     With ``F`` an integer instead of a finite field::
 
-        sage: BD = designs.AffineGeometryDesign(3, 2, 4)                                # optional - sage.rings.finite_rings
-        sage: BD.is_t_design(return_parameters=True)                                    # optional - sage.rings.finite_rings
+        sage: BD = designs.AffineGeometryDesign(3, 2, 4)
+        sage: BD.is_t_design(return_parameters=True)
         (True, (2, 64, 16, 5))
 
     Testing the option ``point_coordinates``::
 
-        sage: designs.AffineGeometryDesign(3, 1, GF(2),                                 # optional - sage.rings.finite_rings
+        sage: designs.AffineGeometryDesign(3, 1, GF(2),
         ....:                              point_coordinates=True).blocks()[0]
         [(0, 0, 0), (0, 0, 1)]
-        sage: designs.AffineGeometryDesign(3, 1, GF(2),                                 # optional - sage.rings.finite_rings
+        sage: designs.AffineGeometryDesign(3, 1, GF(2),
         ....:                              point_coordinates=False).blocks()[0]
         [0, 1]
     """

src/sage/combinat/designs/difference_family.py

@@ -1547,6 +1547,7 @@ def relative_difference_set_from_homomorphism(q, N, d, check=True, return_group=
         return G2, second_diff_set
     return second_diff_set
 
+
 def is_relative_difference_set(R, G, H, params, verbose=False):
     r"""
     Check if ``R`` is a difference set of ``G`` relative to ``H``, with the given parameters.

src/sage/combinat/designs/gen_quadrangles_with_spread.pyx

@@ -1,3 +1,4 @@
+# sage.doctest: optional - sage.libs.gap
 r"""
 Database of generalised quadrangles with spread
 

src/sage/combinat/designs/twographs.py

@@ -291,9 +291,9 @@ def twograph_descendant(G, v, name=None):
         sage: twograph_descendant(T8, v).is_strongly_regular(parameters=True)
         (27, 16, 10, 8)
         sage: p = graphs.PetersenGraph()
-        sage: twograph_descendant(p,5)
+        sage: twograph_descendant(p, 5)
         Graph on 9 vertices
-        sage: twograph_descendant(p,5,name=True)
+        sage: twograph_descendant(p, 5, name=True)
         descendant of Petersen graph at 5: Graph on 9 vertices
     """
     G = G.seidel_switching(G.neighbors(v),inplace=False)