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# 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: _ = designs.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: _ = designs.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: _ = designs.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_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: print 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

`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.incidence_structures.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.

  See also

  - `steiner_triple_system()`
  - `v_4_1_BIBD()`
  - `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, 81, 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, 96, 106, 111, 126, 136, 141]

  But we know some inexistence results:

      sage: designs.balanced_incomplete_block_design(21,6,existence=True)
      False

`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 \(\{0,...,n-1\}\)

  EXAMPLE:

  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.

  See also

  - `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)

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

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

  This method follows the constuction from [ClaytonSmith].

  INPUT:

  - `v` (integer)

  See also

  - `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)

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