Balanced Incomplete Block Designs (BIBD)

This module gathers everything related to Balanced Incomplete Block Designs. One can build a BIBD (or check that it can be built) with balanced_incomplete_block_design():

sage: BIBD = designs.balanced_incomplete_block_design(7,3)

In particular, Sage can build a \((v,k,1)\)-BIBD when one exists for all \(k\leq 5\). The following functions are available:

balanced_incomplete_block_design() Return a BIBD of parameters \(v,k\).
BIBD_from_TD() Return a BIBD through TD-based constructions.
BIBD_from_difference_family() Return the BIBD associated to the difference family D on the group G.
BIBD_from_PBD() Return a \((v,k,1)\)-BIBD from a \((r,K)\)-PBD where \(r=(v-1)/(k-1)\).
PBD_from_TD() Return a \((kt,\{k,t\})\)-PBD if \(u=0\) and a \((kt+u,\{k,k+1,t,u\})\)-PBD otherwise.
steiner_triple_system() Return a Steiner Triple System.
v_5_1_BIBD() Return a \((v,5,1)\)-BIBD.
v_4_1_BIBD() Return a \((v,4,1)\)-BIBD.
PBD_4_5_8_9_12() Return a \((v,\{4,5,8,9,12\})\)-PBD on \(v\) elements.
BIBD_5q_5_for_q_prime_power() Return a \((5q,5,1)\)-BIBD with \(q\equiv 1\pmod 4\) a prime power.

Construction of BIBD when \(k=4\)

Decompositions of \(K_v\) into \(K_4\) (i.e. \((v,4,1)\)-BIBD) are built following Douglas Stinson’s construction as presented in [Stinson2004] page 167. It is based upon the construction of \((v\{4,5,8,9,12\})\)-PBD (see the doc of PBD_4_5_8_9_12()), knowing that a \((v\{4,5,8,9,12\})\)-PBD on \(v\) points can always be transformed into a \(((k-1)v+1,4,1)\)-BIBD, which covers all possible cases of \((v,4,1)\)-BIBD.

Construction of BIBD when \(k=5\)

Decompositions of \(K_v\) into \(K_4\) (i.e. \((v,4,1)\)-BIBD) are built following Clayton Smith’s construction [ClaytonSmith].

[ClaytonSmith](1, 2, 3, 4) On the existence of \((v,5,1)\)-BIBD. http://www.argilo.net/files/bibd.pdf Clayton Smith

Functions

sage.combinat.designs.bibd.BIBD_5q_5_for_q_prime_power(q)

Return a \((5q,5,1)\)-BIBD with \(q\equiv 1\pmod 4\) a prime power.

See Theorem 24 [ClaytonSmith].

INPUT:

  • q (integer) – a prime power such that \(q\equiv 1\pmod 4\).

EXAMPLES:

sage: from sage.combinat.designs.bibd import BIBD_5q_5_for_q_prime_power
sage: for q in [25, 45, 65, 85, 125, 145, 185, 205, 305, 405, 605]: # long time
....:     _ = BIBD_5q_5_for_q_prime_power(q/5)                      # long time
sage.combinat.designs.bibd.BIBD_from_PBD(PBD, v, k, check=True, base_cases={})

Return a \((v,k,1)\)-BIBD from a \((r,K)\)-PBD where \(r=(v-1)/(k-1)\).

This is Theorem 7.20 from [Stinson2004].

INPUT:

  • v,k – integers.
  • PBD – A PBD on \(r=(v-1)/(k-1)\) points, such that for any block of PBD of size \(s\) there must exist a \(((k-1)s+1,k,1)\)-BIBD.
  • 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.
  • base_cases – caching system, for internal use.

EXAMPLES:

sage: from sage.combinat.designs.bibd import PBD_4_5_8_9_12
sage: from sage.combinat.designs.bibd import BIBD_from_PBD
sage: from sage.combinat.designs.bibd import is_pairwise_balanced_design
sage: PBD = PBD_4_5_8_9_12(17)
sage: bibd = is_pairwise_balanced_design(BIBD_from_PBD(PBD,52,4),52,[4])
sage.combinat.designs.bibd.BIBD_from_TD(v, k, existence=False)

Return a BIBD through TD-based constructions.

INPUT:

  • v,k (integers) – computes a \((v,k,1)\)-BIBD.

  • 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 sage.misc.unknown).
    • False – meaning that the design does not exist.

This method implements three constructions:

  • If there exists a \(TD(k,v)\) and a \((v,k,1)\)-BIBD then there exists a \((kv,k,1)\)-BIBD.

    The BIBD is obtained from all blocks of the \(TD\), and from the blocks of the \((v,k,1)\)-BIBDs defined over the \(k\) groups of the \(TD\).

  • If there exists a \(TD(k,v)\) and a \((v+1,k,1)\)-BIBD then there exists a \((kv+1,k,1)\)-BIBD.

    The BIBD is obtained from all blocks of the \(TD\), and from the blocks of the \((v+1,k,1)\)-BIBDs defined over the sets \(V_1\cup \infty,\dots,V_k\cup \infty\) where the \(V_1,\dots,V_k\) are the groups of the TD.

  • If there exists a \(TD(k,v)\) and a \((v+k,k,1)\)-BIBD then there exists a \((kv+k,k,1)\)-BIBD.

    The BIBD is obtained from all blocks of the \(TD\), and from the blocks of the \((v+k,k,1)\)-BIBDs defined over the sets \(V_1\cup \{\infty_1,\dots,\infty_k\},\dots,V_k\cup \{\infty_1,\dots,\infty_k\}\) where the \(V_1,\dots,V_k\) are the groups of the TD. By making sure that all copies of the \((v+k,k,1)\)-BIBD contain the block \(\{\infty_1,\dots,\infty_k\}\), the result is also a BIBD.

These constructions can be found in http://www.argilo.net/files/bibd.pdf.

EXAMPLES:

First construction:

sage: from sage.combinat.designs.bibd import BIBD_from_TD
sage: BIBD_from_TD(25,5,existence=True)
True
sage: _ = BlockDesign(25,BIBD_from_TD(25,5))

Second construction:

sage: from sage.combinat.designs.bibd import BIBD_from_TD
sage: BIBD_from_TD(21,5,existence=True)
True
sage: _ = BlockDesign(21,BIBD_from_TD(21,5))

Third construction:

sage: from sage.combinat.designs.bibd import BIBD_from_TD
sage: BIBD_from_TD(85,5,existence=True)
True
sage: _ = BlockDesign(85,BIBD_from_TD(85,5))

No idea:

sage: from sage.combinat.designs.bibd import BIBD_from_TD
sage: BIBD_from_TD(20,5,existence=True)
Unknown
sage: BIBD_from_TD(20,5)
Traceback (most recent call last):
...
NotImplementedError: I do not know how to build a (20,5,1)-BIBD!
sage.combinat.designs.bibd.BIBD_from_arc_in_desarguesian_projective_plane(n, k, existence=False)

Returns a \((n,k,1)\)-BIBD from a maximal arc in a projective plane.

This function implements a construction from Denniston [Denniston69], who describes a maximal arc in a Desarguesian Projective Plane of order \(2^k\). From two powers of two \(n,q\) with \(n<q\), it produces a \(((n-1)(q+1)+1,n,1)\)-BIBD.

INPUT:

  • n,k (integers) – must be powers of two (among other restrictions).
  • existence (boolean) – whether to return the BIBD obtained through this construction (default), or to merely indicate with a boolean return value whether this method can build the requested BIBD.

EXAMPLES:

A \((232,8,1)\)-BIBD:

sage: from sage.combinat.designs.bibd import BIBD_from_arc_in_desarguesian_projective_plane
sage: from sage.combinat.designs.bibd import BalancedIncompleteBlockDesign
sage: D = BIBD_from_arc_in_desarguesian_projective_plane(232,8)
sage: BalancedIncompleteBlockDesign(232,D)
(232,8,1)-Balanced Incomplete Block Design

A \((120,8,1)\)-BIBD:

sage: D = BIBD_from_arc_in_desarguesian_projective_plane(120,8)
sage: BalancedIncompleteBlockDesign(120,D)
(120,8,1)-Balanced Incomplete Block Design

Other parameters:

sage: all(BIBD_from_arc_in_desarguesian_projective_plane(n,k,existence=True)
....:     for n,k in
....:       [(120, 8), (232, 8), (456, 8), (904, 8), (496, 16),
....:        (976, 16), (1936, 16), (2016, 32), (4000, 32), (8128, 64)])
True

Of course, not all can be built this way:

sage: BIBD_from_arc_in_desarguesian_projective_plane(7,3,existence=True)
False
sage: BIBD_from_arc_in_desarguesian_projective_plane(7,3)
Traceback (most recent call last):
...
ValueError: This function cannot produce a (7,3,1)-BIBD

REFERENCE:

[Denniston69]R. H. F. Denniston, Some maximal arcs in finite projective planes. Journal of Combinatorial Theory 6, no. 3 (1969): 317-319. http://dx.doi.org/10.1016/S0021-9800(69)80095-5
sage.combinat.designs.bibd.BIBD_from_difference_family(G, D, lambd=None, check=True)

Return the BIBD associated to the difference family D on the group G.

Let \(G\) be a group. A \((G,k,\lambda)\)-difference family is a family \(B = \{B_1,B_2,\ldots,B_b\}\) of \(k\)-subsets of \(G\) such that for each element of \(G \backslash \{0\}\) there exists exactly \(\lambda\) pairs of elements \((x,y)\), \(x\) and \(y\) belonging to the same block, such that \(x - y = g\) (or x y^{-1} = g` in multiplicative notation).

If \(\{B_1, B_2, \ldots, B_b\}\) is a \((G,k,\lambda)\)-difference family then its set of translates \(\{B_i \cdot g; i \in \{1,\ldots,b\}, g \in G\}\) is a \((v,k,\lambda)\)-BIBD where \(v\) is the cardinality of \(G\).

INPUT:

  • G - a finite additive Abelian group
  • D - a difference family on G (short blocks are allowed).
  • lambd - the \(\lambda\) parameter (optional, only used if check is True)
  • check - whether or not we check the output (default: True)

EXAMPLES:

sage: G = Zmod(21)
sage: D = [[0,1,4,14,16]]
sage: sorted(G(x-y) for x in D[0] for y in D[0] if x != y)
[1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20]

sage: from sage.combinat.designs.bibd import BIBD_from_difference_family
sage: BIBD_from_difference_family(G, D)
[[0, 1, 4, 14, 16],
 [1, 2, 5, 15, 17],
 [2, 3, 6, 16, 18],
 [3, 4, 7, 17, 19],
 [4, 5, 8, 18, 20],
 [5, 6, 9, 19, 0],
 [6, 7, 10, 20, 1],
 [7, 8, 11, 0, 2],
 [8, 9, 12, 1, 3],
 [9, 10, 13, 2, 4],
 [10, 11, 14, 3, 5],
 [11, 12, 15, 4, 6],
 [12, 13, 16, 5, 7],
 [13, 14, 17, 6, 8],
 [14, 15, 18, 7, 9],
 [15, 16, 19, 8, 10],
 [16, 17, 20, 9, 11],
 [17, 18, 0, 10, 12],
 [18, 19, 1, 11, 13],
 [19, 20, 2, 12, 14],
 [20, 0, 3, 13, 15]]
class sage.combinat.designs.bibd.BalancedIncompleteBlockDesign(points, blocks, k=None, lambd=1, check=True, copy=True, **kwds)

Bases: sage.combinat.designs.bibd.PairwiseBalancedDesign

” Balanced Incomplete Block Design (BIBD)

INPUT:

  • points – the underlying set. If points is an integer \(v\), then the set is considered to be \(\{0, ..., v-1\}\).
  • blocks – collection of blocks
  • k (integer) – size of the blocks. Set to None (automatic guess) by default.
  • lambd (integer) – value of \(\lambda\), set to \(1\) by default.
  • check (boolean) – whether to check that the design is a \(PBD\) with the right parameters.
  • copy – (use with caution) if set to False then blocks must be a list of lists of integers. The list will not be copied but will be modified in place (each block is sorted, and the whole list is sorted). Your blocks object will become the instance’s internal data.

EXAMPLES:

sage: b=designs.balanced_incomplete_block_design(9,3); b
(9,3,1)-Balanced Incomplete Block Design
arc(s=2, solver=None, verbose=0)

Return the s-arc with maximum cardinality.

A \(s\)-arc is a subset of points in a BIBD that intersects each block on at most \(s\) points. It is one possible generalization of independent set for graphs.

A simple counting shows that the cardinality of a \(s\)-arc is at most \((s-1) * r + 1\) where \(r\) is the number of blocks incident to any point. A \(s\)-arc in a BIBD with cardinality \((s-1) * r + 1\) is called maximal and is characterized by the following property: it is not empty and each block either contains \(0\) or \(s\) points of this arc. Equivalently, the trace of the BIBD on these points is again a BIBD (with block size \(s\)).

For more informations, see Wikipedia article Arc_(projective_geometry).

INPUT:

  • s - (default to 2) the maximum number of points from the arc in each block
  • solver – (default: None) Specify a Linear Program (LP) solver to be used. If set to None, the default one is used. For more information on LP solvers and which default solver is used, see the method solve of the class MixedIntegerLinearProgram.
  • verbose – integer (default: 0). Sets the level of verbosity. Set to 0 by default, which means quiet.

EXAMPLES:

sage: B = designs.balanced_incomplete_block_design(21, 5)
sage: a2 = B.arc()
sage: a2 # random
[5, 9, 10, 12, 15, 20]
sage: len(a2)
6
sage: a4 = B.arc(4)
sage: a4 # random
[0, 1, 2, 5, 6, 8, 9, 10, 11, 12, 13, 14, 15, 16, 18, 20]
sage: len(a4)
16

The \(2\)-arc and \(4\)-arc above are maximal. One can check that they intersect the blocks in either 0 or \(s\) points. Or equivalently that the traces are again BIBD:

sage: r = (21-1)//(5-1)
sage: 1 + r*1
6
sage: 1 + r*3
16

sage: B.trace(a2).is_t_design(2, return_parameters=True)
(True, (2, 6, 2, 1))
sage: B.trace(a4).is_t_design(2, return_parameters=True)
(True, (2, 16, 4, 1))

Some other examples which are not maximal:

sage: B = designs.balanced_incomplete_block_design(25, 4)
sage: a2 = B.arc(2)
sage: r = (25-1)//(4-1)
sage: len(a2), 1 + r
(8, 9)
sage: sa2 = set(a2)
sage: set(len(sa2.intersection(b)) for b in B.blocks())
{0, 1, 2}
sage: B.trace(a2).is_t_design(2)
False

sage: a3 = B.arc(3)
sage: len(a3), 1 + 2*r
(15, 17)
sage: sa3 = set(a3)
sage: set(len(sa3.intersection(b)) for b in B.blocks()) == set([0,3])
False
sage: B.trace(a3).is_t_design(3)
False
sage.combinat.designs.bibd.PBD_4_5_8_9_12(v, check=True)

Return a \((v,\{4,5,8,9,12\})\)-PBD on \(v\) elements.

A \((v,\{4,5,8,9,12\})\)-PBD exists if and only if \(v\equiv 0,1 \pmod 4\). The construction implemented here appears page 168 in [Stinson2004].

INPUT:

  • v – an integer congruent to \(0\) or \(1\) modulo \(4\).
  • 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: designs.balanced_incomplete_block_design(40,4).blocks() # indirect doctest
[[0, 1, 2, 12], [0, 3, 6, 9], [0, 4, 8, 10],
 [0, 5, 7, 11], [0, 13, 26, 39], [0, 14, 25, 28],
 [0, 15, 27, 38], [0, 16, 22, 32], [0, 17, 23, 34],
...

Check that trac ticket #16476 is fixed:

sage: from sage.combinat.designs.bibd import PBD_4_5_8_9_12
sage: for v in (0,1,4,5,8,9,12,13,16,17,20,21,24,25):
....:     _ = PBD_4_5_8_9_12(v)
sage.combinat.designs.bibd.PBD_from_TD(k, t, u)

Return a \((kt,\{k,t\})\)-PBD if \(u=0\) and a \((kt+u,\{k,k+1,t,u\})\)-PBD otherwise.

This is theorem 23 from [ClaytonSmith]. The PBD is obtained from the blocks a truncated \(TD(k+1,t)\), to which are added the blocks corresponding to the groups of the TD. When \(u=0\), a \(TD(k,t)\) is used instead.

INPUT:

  • k,t,u – integers such that \(0\leq u \leq t\).

EXAMPLES:

sage: from sage.combinat.designs.bibd import PBD_from_TD
sage: from sage.combinat.designs.bibd import is_pairwise_balanced_design
sage: PBD = PBD_from_TD(2,2,1); PBD
[[0, 2, 4], [0, 3], [1, 2], [1, 3, 4], [0, 1], [2, 3]]
sage: is_pairwise_balanced_design(PBD,2*2+1,[2,3])
True
class sage.combinat.designs.bibd.PairwiseBalancedDesign(points, blocks, K=None, lambd=1, check=True, copy=True, **kwds)

Bases: sage.combinat.designs.group_divisible_designs.GroupDivisibleDesign

Pairwise Balanced Design (PBD)

A Pairwise Balanced Design, or \((v,K,\lambda)\)-PBD, is a collection \(\mathcal B\) of blocks defined on a set \(X\) of size \(v\), such that any block pair of points \(p_1,p_2\in X\) occurs in exactly \(\lambda\) blocks of \(\mathcal B\). Besides, for every block \(B\in \mathcal B\) we must have \(|B|\in K\).

INPUT:

  • points – the underlying set. If points is an integer \(v\), then the set is considered to be \(\{0, ..., v-1\}\).
  • blocks – collection of blocks
  • K – list of integers of which the sizes of the blocks must be elements. Set to None (automatic guess) by default.
  • lambd (integer) – value of \(\lambda\), set to \(1\) by default.
  • check (boolean) – whether to check that the design is a \(PBD\) with the right parameters.
  • copy – (use with caution) if set to False then blocks must be a list of lists of integers. The list will not be copied but will be modified in place (each block is sorted, and the whole list is sorted). Your blocks object will become the instance’s internal data.
sage.combinat.designs.bibd.balanced_incomplete_block_design(v, k, existence=False, use_LJCR=False)

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 corresponding Wikipedia entry.

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 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 best_known_covering_design_www()). This requires internet.

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!
sage.combinat.designs.bibd.steiner_triple_system(n)

Return a Steiner Triple System

A Steiner Triple System (STS) of a set \(\{0,...,n-1\}\) is a family \(S\) of 3-sets such that for any \(i \not = j\) there exists exactly one set of \(S\) in which they are both contained.

It can alternatively be thought of as a factorization of the complete graph \(K_n\) with triangles.

A Steiner Triple System of a \(n\)-set exists if and only if \(n \equiv 1 \pmod 6\) or \(n \equiv 3 \pmod 6\), in which case one can be found through Bose’s and Skolem’s constructions, respectively [AndHonk97].

INPUT:

  • n return a Steiner Triple System of \(\{0,...,n-1\}\)

EXAMPLES:

A Steiner Triple System on \(9\) elements

sage: sts = designs.steiner_triple_system(9)
sage: sts
(9,3,1)-Balanced Incomplete Block Design
sage: list(sts)
[[0, 1, 5], [0, 2, 4], [0, 3, 6], [0, 7, 8], [1, 2, 3],
 [1, 4, 7], [1, 6, 8], [2, 5, 8], [2, 6, 7], [3, 4, 8],
 [3, 5, 7], [4, 5, 6]]

As any pair of vertices is covered once, its parameters are

sage: sts.is_t_design(return_parameters=True)
(True, (2, 9, 3, 1))

An exception is raised for invalid values of n

sage: designs.steiner_triple_system(10)
Traceback (most recent call last):
...
EmptySetError: Steiner triple systems only exist for n = 1 mod 6 or n = 3 mod 6

REFERENCE:

[AndHonk97]A short course in Combinatorial Designs, Ian Anderson, Iiro Honkala, Internet Editions, Spring 1997, http://www.utu.fi/~honkala/designs.ps
sage.combinat.designs.bibd.v_4_1_BIBD(v, check=True)

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 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.

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
sage.combinat.designs.bibd.v_5_1_BIBD(v, check=True)

Return a \((v,5,1)\)-BIBD.

This method follows the construction from [ClaytonSmith].

INPUT:

  • v (integer)

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)