def v_5_1_BIBD(v, check=True): r""" Return a `(v,5,1)`-BIBD. This method follows the constuction from [ClaytonSmith]_. INPUT: - ``v`` (integer) .. SEEALSO:: * :func:`balanced_incomplete_block_design` EXAMPLES:: sage: from sage.combinat.designs.bibd import v_5_1_BIBD sage: i = 0 sage: while i<200: ....: i += 20 ....: _ = v_5_1_BIBD(i+1) ....: _ = v_5_1_BIBD(i+5) TESTS: Check that the needed difference families are there:: sage: for v in [21,41,61,81,141,161,281]: ....: assert designs.difference_family(v,5,existence=True) ....: _ = designs.difference_family(v,5) """ v = int(v) assert (v > 1) assert (v%20 == 5 or v%20 == 1) # note: equivalent to (v-1)%4 == 0 and (v*(v-1))%20 == 0 # Lemma 27 if v%5 == 0 and (v//5)%4 == 1 and is_prime_power(v//5): bibd = BIBD_5q_5_for_q_prime_power(v//5) # Lemma 28 elif v in [21,41,61,81,141,161,281]: from difference_family import difference_family G,D = difference_family(v,5) bibd = BIBD_from_difference_family(G, D, check=False) # Lemma 29 elif v == 165: bibd = BIBD_from_PBD(v_5_1_BIBD(41,check=False),165,5,check=False) elif v == 181: bibd = BIBD_from_PBD(v_5_1_BIBD(45,check=False),181,5,check=False) elif v in (201,285,301,401,421,425): # Call directly the BIBD_from_TD function # note: there are (201,5,1) and (421,5)-difference families that can be # obtained from the general constructor bibd = BIBD_from_TD(v,5) # Theorem 31.2 elif (v-1)//4 in [80, 81, 85, 86, 90, 91, 95, 96, 110, 111, 115, 116, 120, 121, 250, 251, 255, 256, 260, 261, 265, 266, 270, 271]: r = (v-1)//4 if r <= 96: k,t,u = 5, 16, r-80 elif r <= 121: k,t,u = 10, 11, r-110 else: k,t,u = 10, 25, r-250 bibd = BIBD_from_PBD(PBD_from_TD(k,t,u),v,5,check=False) else: r,s,t,u = _get_r_s_t_u(v) bibd = BIBD_from_PBD(PBD_from_TD(5,t,u),v,5,check=False) if check: assert is_pairwise_balanced_design(bibd,v,[5]) return bibd
def balanced_incomplete_block_design(v, k, existence=False, use_LJCR=False): r""" Return a BIBD of parameters `v,k`. A Balanced Incomplete Block Design of parameters `v,k` is a collection `\mathcal C` of `k`-subsets of `V=\{0,\dots,v-1\}` such that for any two distinct elements `x,y\in V` there is a unique element `S\in \mathcal C` such that `x,y\in S`. More general definitions sometimes involve a `\lambda` parameter, and we assume here that `\lambda=1`. For more information on BIBD, see the :wikipedia:`corresponding Wikipedia entry <Block_design#Definition_of_a_BIBD_.28or_2-design.29>`. INPUT: - ``v,k`` (integers) - ``existence`` (boolean) -- instead of building the design, return: - ``True`` -- meaning that Sage knows how to build the design - ``Unknown`` -- meaning that Sage does not know how to build the design, but that the design may exist (see :mod:`sage.misc.unknown`). - ``False`` -- meaning that the design does not exist. - ``use_LJCR`` (boolean) -- whether to query the La Jolla Covering Repository for the design when Sage does not know how to build it (see :func:`~sage.combinat.designs.covering_design.best_known_covering_design_www`). This requires internet. .. SEEALSO:: * :func:`steiner_triple_system` * :func:`v_4_1_BIBD` * :func:`v_5_1_BIBD` TODO: * Implement other constructions from the Handbook of Combinatorial Designs. EXAMPLES:: sage: designs.balanced_incomplete_block_design(7, 3).blocks() [[0, 1, 3], [0, 2, 4], [0, 5, 6], [1, 2, 6], [1, 4, 5], [2, 3, 5], [3, 4, 6]] sage: B = designs.balanced_incomplete_block_design(66, 6, use_LJCR=True) # optional - internet sage: B # optional - internet Incidence structure with 66 points and 143 blocks sage: B.blocks() # optional - internet [[0, 1, 2, 3, 4, 65], [0, 5, 24, 25, 39, 57], [0, 6, 27, 38, 44, 55], ... sage: designs.balanced_incomplete_block_design(66, 6, use_LJCR=True) # optional - internet Incidence structure with 66 points and 143 blocks sage: designs.balanced_incomplete_block_design(216, 6) Traceback (most recent call last): ... NotImplementedError: I don't know how to build a (216,6,1)-BIBD! TESTS:: sage: designs.balanced_incomplete_block_design(85,5,existence=True) True sage: _ = designs.balanced_incomplete_block_design(85,5) A BIBD from a Finite Projective Plane:: sage: _ = designs.balanced_incomplete_block_design(21,5) Some trivial BIBD:: sage: designs.balanced_incomplete_block_design(10,10) (10,10,1)-Balanced Incomplete Block Design sage: designs.balanced_incomplete_block_design(1,10) (1,0,1)-Balanced Incomplete Block Design Existence of BIBD with `k=3,4,5`:: sage: [v for v in xrange(50) if designs.balanced_incomplete_block_design(v,3,existence=True)] [1, 3, 7, 9, 13, 15, 19, 21, 25, 27, 31, 33, 37, 39, 43, 45, 49] sage: [v for v in xrange(100) if designs.balanced_incomplete_block_design(v,4,existence=True)] [1, 4, 13, 16, 25, 28, 37, 40, 49, 52, 61, 64, 73, 76, 85, 88, 97] sage: [v for v in xrange(150) if designs.balanced_incomplete_block_design(v,5,existence=True)] [1, 5, 21, 25, 41, 45, 61, 65, 81, 85, 101, 105, 121, 125, 141, 145] For `k > 5` there are currently very few constructions:: sage: [v for v in xrange(300) if designs.balanced_incomplete_block_design(v,6,existence=True) is True] [1, 6, 31, 66, 76, 91, 96, 106, 111, 121, 126, 136, 141, 151, 156, 171, 181, 186, 196, 201, 211, 241, 271] sage: [v for v in xrange(300) if designs.balanced_incomplete_block_design(v,6,existence=True) is Unknown] [51, 61, 81, 166, 216, 226, 231, 246, 256, 261, 276, 286, 291] Here are some constructions with `k \geq 7` and `v` a prime power:: sage: designs.balanced_incomplete_block_design(169,7) (169,7,1)-Balanced Incomplete Block Design sage: designs.balanced_incomplete_block_design(617,8) (617,8,1)-Balanced Incomplete Block Design sage: designs.balanced_incomplete_block_design(433,9) (433,9,1)-Balanced Incomplete Block Design sage: designs.balanced_incomplete_block_design(1171,10) (1171,10,1)-Balanced Incomplete Block Design And we know some inexistence results:: sage: designs.balanced_incomplete_block_design(21,6,existence=True) False """ lmbd = 1 # Trivial BIBD if v == 1: if existence: return True return BalancedIncompleteBlockDesign(v, [], check=False) if k == v: if existence: return True return BalancedIncompleteBlockDesign(v, [range(v)], check=False, copy=False) # Non-existence of BIBD if (v < k or k < 2 or (v-1) % (k-1) != 0 or (v*(v-1)) % (k*(k-1)) != 0 or # From the Handbook of combinatorial designs: # # With lambda>1 other exceptions are # (15,5,2),(21,6,2),(22,7,2),(22,8,4). (k==6 and v in [36,46]) or (k==7 and v == 43) or # Fisher's inequality (v*(v-1))/(k*(k-1)) < v): if existence: return False raise EmptySetError("There exists no ({},{},{})-BIBD".format(v,k,lmbd)) if k == 2: if existence: return True from itertools import combinations return BalancedIncompleteBlockDesign(v, combinations(range(v),2), check=False, copy=True) if k == 3: if existence: return v%6 == 1 or v%6 == 3 return steiner_triple_system(v) if k == 4: if existence: return v%12 == 1 or v%12 == 4 return BalancedIncompleteBlockDesign(v, v_4_1_BIBD(v), copy=False) if k == 5: if existence: return v%20 == 1 or v%20 == 5 return BalancedIncompleteBlockDesign(v, v_5_1_BIBD(v), copy=False) from difference_family import difference_family from database import BIBD_constructions if (v,k,1) in BIBD_constructions: if existence: return True return BlockDesign(v,BIBD_constructions[(v,k,1)](), copy=False) if BIBD_from_arc_in_desarguesian_projective_plane(v,k,existence=True): if existence: return True B = BIBD_from_arc_in_desarguesian_projective_plane(v,k) return BalancedIncompleteBlockDesign(v, B, copy=False) if BIBD_from_TD(v,k,existence=True): if existence: return True return BalancedIncompleteBlockDesign(v, BIBD_from_TD(v,k), copy=False) if v == (k-1)**2+k and is_prime_power(k-1): if existence: return True from block_design import projective_plane return BalancedIncompleteBlockDesign(v, projective_plane(k-1),copy=False) if difference_family(v,k,existence=True): if existence: return True G,D = difference_family(v,k) return BalancedIncompleteBlockDesign(v, BIBD_from_difference_family(G,D,check=False), copy=False) if use_LJCR: from covering_design import best_known_covering_design_www B = best_known_covering_design_www(v,k,2) # Is it a BIBD or just a good covering ? expected_n_of_blocks = binomial(v,2)/binomial(k,2) if B.low_bd() > expected_n_of_blocks: if existence: return False raise EmptySetError("There exists no ({},{},{})-BIBD".format(v,k,lmbd)) B = B.incidence_structure() if B.num_blocks() == expected_n_of_blocks: if existence: return True else: return B if existence: return Unknown else: raise NotImplementedError("I don't know how to build a ({},{},1)-BIBD!".format(v,k))
def v_4_1_BIBD(v, check=True): r""" Return a `(v,4,1)`-BIBD. A `(v,4,1)`-BIBD is an edge-decomposition of the complete graph `K_v` into copies of `K_4`. For more information, see :func:`balanced_incomplete_block_design`. It exists if and only if `v\equiv 1,4 \pmod {12}`. See page 167 of [Stinson2004]_ for the construction details. .. SEEALSO:: * :func:`balanced_incomplete_block_design` INPUT: - ``v`` (integer) -- number of points. - ``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.designs.bibd import v_4_1_BIBD # long time sage: for n in range(13,100): # long time ....: if n%12 in [1,4]: # long time ....: _ = v_4_1_BIBD(n, check = True) # long time TESTS: Check that the `(25,4)` and `(37,4)`-difference family are available:: sage: assert designs.difference_family(25,4,existence=True) sage: _ = designs.difference_family(25,4) sage: assert designs.difference_family(37,4,existence=True) sage: _ = designs.difference_family(37,4) Check some larger `(v,4,1)`-BIBD (see :trac:`17557`):: sage: for v in range(400): # long time ....: if v%12 in [1,4]: # long time ....: _ = designs.balanced_incomplete_block_design(v,4) # long time """ k = 4 if v == 0: return [] if v <= 12 or v%12 not in [1,4]: raise EmptySetError("A K_4-decomposition of K_v exists iif v=2,4 mod 12, v>12 or v==0") # Step 1. Base cases. if v == 13: # note: this construction can also be obtained from difference_family from block_design import projective_plane return projective_plane(3)._blocks if v == 16: from block_design import AffineGeometryDesign from sage.rings.finite_rings.constructor import FiniteField return AffineGeometryDesign(2,1,FiniteField(4,'x'))._blocks if v == 25 or v == 37: from difference_family import difference_family G,D = difference_family(v,4) return BIBD_from_difference_family(G,D,check=False) if v == 28: return [[0, 1, 23, 26], [0, 2, 10, 11], [0, 3, 16, 18], [0, 4, 15, 20], [0, 5, 8, 9], [0, 6, 22, 25], [0, 7, 14, 21], [0, 12, 17, 27], [0, 13, 19, 24], [1, 2, 24, 27], [1, 3, 11, 12], [1, 4, 17, 19], [1, 5, 14, 16], [1, 6, 9, 10], [1, 7, 20, 25], [1, 8, 15, 22], [1, 13, 18, 21], [2, 3, 21, 25], [2, 4, 12, 13], [2, 5, 18, 20], [2, 6, 15, 17], [2, 7, 19, 22], [2, 8, 14, 26], [2, 9, 16, 23], [3, 4, 22, 26], [3, 5, 7, 13], [3, 6, 14, 19], [3, 8, 20, 23], [3, 9, 15, 27], [3, 10, 17, 24], [4, 5, 23, 27], [4, 6, 7, 8], [4, 9, 14, 24], [4, 10, 16, 21], [4, 11, 18, 25], [5, 6, 21, 24], [5, 10, 15, 25], [5, 11, 17, 22], [5, 12, 19, 26], [6, 11, 16, 26], [6, 12, 18, 23], [6, 13, 20, 27], [7, 9, 17, 18], [7, 10, 26, 27], [7, 11, 23, 24], [7, 12, 15, 16], [8, 10, 18, 19], [8, 11, 21, 27], [8, 12, 24, 25], [8, 13, 16, 17], [9, 11, 19, 20], [9, 12, 21, 22], [9, 13, 25, 26], [10, 12, 14, 20], [10, 13, 22, 23], [11, 13, 14, 15], [14, 17, 23, 25], [14, 18, 22, 27], [15, 18, 24, 26], [15, 19, 21, 23], [16, 19, 25, 27], [16, 20, 22, 24], [17, 20, 21, 26]] # Step 2 : this is function PBD_4_5_8_9_12 PBD = PBD_4_5_8_9_12((v-1)/(k-1),check=False) # Step 3 : Theorem 7.20 bibd = BIBD_from_PBD(PBD,v,k,check=False) if check: assert is_pairwise_balanced_design(bibd,v,[k]) return bibd
def balanced_incomplete_block_design(v, k, existence=False, use_LJCR=False): r""" Return a BIBD of parameters `v,k`. A Balanced Incomplete Block Design of parameters `v,k` is a collection `\mathcal C` of `k`-subsets of `V=\{0,\dots,v-1\}` such that for any two distinct elements `x,y\in V` there is a unique element `S\in \mathcal C` such that `x,y\in S`. More general definitions sometimes involve a `\lambda` parameter, and we assume here that `\lambda=1`. For more information on BIBD, see the :wikipedia:`corresponding Wikipedia entry <Block_design#Definition_of_a_BIBD_.28or_2-design.29>`. INPUT: - ``v,k`` (integers) - ``existence`` (boolean) -- instead of building the design, return: - ``True`` -- meaning that Sage knows how to build the design - ``Unknown`` -- meaning that Sage does not know how to build the design, but that the design may exist (see :mod:`sage.misc.unknown`). - ``False`` -- meaning that the design does not exist. - ``use_LJCR`` (boolean) -- whether to query the La Jolla Covering Repository for the design when Sage does not know how to build it (see :func:`~sage.combinat.designs.covering_design.best_known_covering_design_www`). This requires internet. .. SEEALSO:: * :func:`steiner_triple_system` * :func:`v_4_1_BIBD` * :func:`v_5_1_BIBD` TODO: * Implement other constructions from the Handbook of Combinatorial Designs. EXAMPLES:: sage: designs.balanced_incomplete_block_design(7, 3).blocks() [[0, 1, 3], [0, 2, 4], [0, 5, 6], [1, 2, 6], [1, 4, 5], [2, 3, 5], [3, 4, 6]] sage: B = designs.balanced_incomplete_block_design(66, 6, use_LJCR=True) # optional - internet sage: B # optional - internet Incidence structure with 66 points and 143 blocks sage: B.blocks() # optional - internet [[0, 1, 2, 3, 4, 65], [0, 5, 24, 25, 39, 57], [0, 6, 27, 38, 44, 55], ... sage: designs.balanced_incomplete_block_design(66, 6, use_LJCR=True) # optional - internet Incidence structure with 66 points and 143 blocks sage: designs.balanced_incomplete_block_design(141, 6) Traceback (most recent call last): ... NotImplementedError: I don't know how to build a (141,6,1)-BIBD! TESTS:: sage: designs.balanced_incomplete_block_design(85,5,existence=True) True sage: _ = designs.balanced_incomplete_block_design(85,5) A BIBD from a Finite Projective Plane:: sage: _ = designs.balanced_incomplete_block_design(21,5) Some trivial BIBD:: sage: designs.balanced_incomplete_block_design(10,10) (10,10,1)-Balanced Incomplete Block Design sage: designs.balanced_incomplete_block_design(1,10) (1,0,1)-Balanced Incomplete Block Design Existence of BIBD with `k=3,4,5`:: sage: [v for v in xrange(50) if designs.balanced_incomplete_block_design(v,3,existence=True)] [1, 3, 7, 9, 13, 15, 19, 21, 25, 27, 31, 33, 37, 39, 43, 45, 49] sage: [v for v in xrange(100) if designs.balanced_incomplete_block_design(v,4,existence=True)] [1, 4, 13, 16, 25, 28, 37, 40, 49, 52, 61, 64, 73, 76, 85, 88, 97] sage: [v for v in xrange(150) if designs.balanced_incomplete_block_design(v,5,existence=True)] [1, 5, 21, 25, 41, 45, 61, 65, 81, 85, 101, 105, 121, 125, 141, 145] For `k > 5` there are currently very few constructions:: sage: [v for v in xrange(150) if designs.balanced_incomplete_block_design(v,6,existence=True) is True] [1, 6, 31, 91, 121] sage: [v for v in xrange(150) if designs.balanced_incomplete_block_design(v,6,existence=True) is Unknown] [51, 61, 66, 76, 81, 96, 106, 111, 126, 136, 141] Here are some constructions with `k \geq 7` and `v` a prime power:: sage: designs.balanced_incomplete_block_design(169,7) (169,7,1)-Balanced Incomplete Block Design sage: designs.balanced_incomplete_block_design(617,8) (617,8,1)-Balanced Incomplete Block Design sage: designs.balanced_incomplete_block_design(433,9) (433,9,1)-Balanced Incomplete Block Design sage: designs.balanced_incomplete_block_design(1171,10) (1171,10,1)-Balanced Incomplete Block Design And we know some inexistence results:: sage: designs.balanced_incomplete_block_design(21,6,existence=True) False """ lmbd = 1 # Trivial BIBD if v == 1: if existence: return True return BalancedIncompleteBlockDesign(v, [], check=False) if k == v: if existence: return True return BalancedIncompleteBlockDesign(v, [range(v)], check=False, copy=False) # Non-existence of BIBD if (v < k or k < 2 or (v - 1) % (k - 1) != 0 or (v * (v - 1)) % (k * (k - 1)) != 0 or # From the Handbook of combinatorial designs: # # With lambda>1 other exceptions are # (15,5,2),(21,6,2),(22,7,2),(22,8,4). (k == 6 and v in [36, 46]) or (k == 7 and v == 43) or # Fisher's inequality (v * (v - 1)) / (k * (k - 1)) < v): if existence: return False raise EmptySetError("There exists no ({},{},{})-BIBD".format( v, k, lmbd)) if k == 2: if existence: return True from itertools import combinations return BalancedIncompleteBlockDesign(v, combinations(range(v), 2), check=False, copy=True) if k == 3: if existence: return v % 6 == 1 or v % 6 == 3 return steiner_triple_system(v) if k == 4: if existence: return v % 12 == 1 or v % 12 == 4 return BalancedIncompleteBlockDesign(v, v_4_1_BIBD(v), copy=False) if k == 5: if existence: return v % 20 == 1 or v % 20 == 5 return BalancedIncompleteBlockDesign(v, v_5_1_BIBD(v), copy=False) from difference_family import difference_family from database import BIBD_constructions if (v, k, 1) in BIBD_constructions: if existence: return True return BlockDesign(v, BIBD_constructions[(v, k, 1)](), copy=False) if BIBD_from_TD(v, k, existence=True): if existence: return True return BalancedIncompleteBlockDesign(v, BIBD_from_TD(v, k), copy=False) if v == (k - 1)**2 + k and is_prime_power(k - 1): if existence: return True from block_design import projective_plane return BalancedIncompleteBlockDesign(v, projective_plane(k - 1), copy=False) if difference_family(v, k, existence=True): if existence: return True G, D = difference_family(v, k) return BalancedIncompleteBlockDesign(v, BIBD_from_difference_family( G, D, check=False), copy=False) if use_LJCR: from covering_design import best_known_covering_design_www B = best_known_covering_design_www(v, k, 2) # Is it a BIBD or just a good covering ? expected_n_of_blocks = binomial(v, 2) / binomial(k, 2) if B.low_bd() > expected_n_of_blocks: if existence: return False raise EmptySetError("There exists no ({},{},{})-BIBD".format( v, k, lmbd)) B = B.incidence_structure() if B.num_blocks() == expected_n_of_blocks: if existence: return True else: return B if existence: return Unknown else: raise NotImplementedError( "I don't know how to build a ({},{},1)-BIBD!".format(v, k))
def v_4_1_BIBD(v, check=True): r""" Return a `(v,4,1)`-BIBD. A `(v,4,1)`-BIBD is an edge-decomposition of the complete graph `K_v` into copies of `K_4`. For more information, see :func:`balanced_incomplete_block_design`. It exists if and only if `v\equiv 1,4 \pmod {12}`. See page 167 of [Stinson2004]_ for the construction details. .. SEEALSO:: * :func:`balanced_incomplete_block_design` INPUT: - ``v`` (integer) -- number of points. - ``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.designs.bibd import v_4_1_BIBD # long time sage: for n in range(13,100): # long time ....: if n%12 in [1,4]: # long time ....: _ = v_4_1_BIBD(n, check = True) # long time TESTS: Check that the `(25,4)` and `(37,4)`-difference family are available:: sage: assert designs.difference_family(25,4,existence=True) sage: _ = designs.difference_family(25,4) sage: assert designs.difference_family(37,4,existence=True) sage: _ = designs.difference_family(37,4) Check some larger `(v,4,1)`-BIBD (see :trac:`17557`):: sage: for v in range(400): # long time ....: if v%12 in [1,4]: # long time ....: _ = designs.balanced_incomplete_block_design(v,4) # long time """ k = 4 if v == 0: return [] if v <= 12 or v % 12 not in [1, 4]: raise EmptySetError( "A K_4-decomposition of K_v exists iif v=2,4 mod 12, v>12 or v==0") # Step 1. Base cases. if v == 13: # note: this construction can also be obtained from difference_family from block_design import projective_plane return projective_plane(3)._blocks if v == 16: from block_design import AffineGeometryDesign from sage.rings.finite_rings.constructor import FiniteField return AffineGeometryDesign(2, 1, FiniteField(4, 'x'))._blocks if v == 25 or v == 37: from difference_family import difference_family G, D = difference_family(v, 4) return BIBD_from_difference_family(G, D, check=False) if v == 28: return [[0, 1, 23, 26], [0, 2, 10, 11], [0, 3, 16, 18], [0, 4, 15, 20], [0, 5, 8, 9], [0, 6, 22, 25], [0, 7, 14, 21], [0, 12, 17, 27], [0, 13, 19, 24], [1, 2, 24, 27], [1, 3, 11, 12], [1, 4, 17, 19], [1, 5, 14, 16], [1, 6, 9, 10], [1, 7, 20, 25], [1, 8, 15, 22], [1, 13, 18, 21], [2, 3, 21, 25], [2, 4, 12, 13], [2, 5, 18, 20], [2, 6, 15, 17], [2, 7, 19, 22], [2, 8, 14, 26], [2, 9, 16, 23], [3, 4, 22, 26], [3, 5, 7, 13], [3, 6, 14, 19], [3, 8, 20, 23], [3, 9, 15, 27], [3, 10, 17, 24], [4, 5, 23, 27], [4, 6, 7, 8], [4, 9, 14, 24], [4, 10, 16, 21], [4, 11, 18, 25], [5, 6, 21, 24], [5, 10, 15, 25], [5, 11, 17, 22], [5, 12, 19, 26], [6, 11, 16, 26], [6, 12, 18, 23], [6, 13, 20, 27], [7, 9, 17, 18], [7, 10, 26, 27], [7, 11, 23, 24], [7, 12, 15, 16], [8, 10, 18, 19], [8, 11, 21, 27], [8, 12, 24, 25], [8, 13, 16, 17], [9, 11, 19, 20], [9, 12, 21, 22], [9, 13, 25, 26], [10, 12, 14, 20], [10, 13, 22, 23], [11, 13, 14, 15], [14, 17, 23, 25], [14, 18, 22, 27], [15, 18, 24, 26], [15, 19, 21, 23], [16, 19, 25, 27], [16, 20, 22, 24], [17, 20, 21, 26]] # Step 2 : this is function PBD_4_5_8_9_12 PBD = PBD_4_5_8_9_12((v - 1) / (k - 1), check=False) # Step 3 : Theorem 7.20 bibd = BIBD_from_PBD(PBD, v, k, check=False) if check: assert is_pairwise_balanced_design(bibd, v, [k]) return bibd
def v_5_1_BIBD(v, check=True): r""" Return a `(v,5,1)`-BIBD. This method follows the constuction from [ClaytonSmith]_. INPUT: - ``v`` (integer) .. SEEALSO:: * :func:`balanced_incomplete_block_design` EXAMPLES:: sage: from sage.combinat.designs.bibd import v_5_1_BIBD sage: i = 0 sage: while i<200: ....: i += 20 ....: _ = v_5_1_BIBD(i+1) ....: _ = v_5_1_BIBD(i+5) TESTS: Check that the needed difference families are there:: sage: for v in [21,41,61,81,141,161,281]: ....: assert designs.difference_family(v,5,existence=True) ....: _ = designs.difference_family(v,5) """ v = int(v) assert (v > 1) assert (v % 20 == 5 or v % 20 == 1 ) # note: equivalent to (v-1)%4 == 0 and (v*(v-1))%20 == 0 # Lemma 27 if v % 5 == 0 and (v // 5) % 4 == 1 and is_prime_power(v // 5): bibd = BIBD_5q_5_for_q_prime_power(v // 5) # Lemma 28 elif v in [21, 41, 61, 81, 141, 161, 281]: from difference_family import difference_family G, D = difference_family(v, 5) bibd = BIBD_from_difference_family(G, D, check=False) # Lemma 29 elif v == 165: bibd = BIBD_from_PBD(v_5_1_BIBD(41, check=False), 165, 5, check=False) elif v == 181: bibd = BIBD_from_PBD(v_5_1_BIBD(45, check=False), 181, 5, check=False) elif v in (201, 285, 301, 401, 421, 425): # Call directly the BIBD_from_TD function # note: there are (201,5,1) and (421,5)-difference families that can be # obtained from the general constructor bibd = BIBD_from_TD(v, 5) # Theorem 31.2 elif (v - 1) // 4 in [ 80, 81, 85, 86, 90, 91, 95, 96, 110, 111, 115, 116, 120, 121, 250, 251, 255, 256, 260, 261, 265, 266, 270, 271 ]: r = (v - 1) // 4 if r <= 96: k, t, u = 5, 16, r - 80 elif r <= 121: k, t, u = 10, 11, r - 110 else: k, t, u = 10, 25, r - 250 bibd = BIBD_from_PBD(PBD_from_TD(k, t, u), v, 5, check=False) else: r, s, t, u = _get_r_s_t_u(v) bibd = BIBD_from_PBD(PBD_from_TD(5, t, u), v, 5, check=False) if check: assert is_pairwise_balanced_design(bibd, v, [5]) return bibd