Group-Divisible Designs (GDD)

This module gathers everything related to Group-Divisible Designs. The constructions defined here can be accessed through designs.<tab>:

sage: designs.group_divisible_design(14,{4},{2})
Group Divisible Design on 14 points of type 2^7

The main function implemented here is group_divisible_design() (which calls all others) and the main class is GroupDivisibleDesign. The following functions are available:

group_divisible_design() Return a \((v,K,G)\)-Group Divisible Design.
GDD_4_2() Return a \((2q,\{4\},\{2\})\)-GDD for \(q\) a prime power with \(q\equiv 1\pmod{6}\).

Functions

sage.combinat.designs.group_divisible_designs.GDD_4_2(q, existence=False, check=True)

Return a \((2q,\{4\},\{2\})\)-GDD for \(q\) a prime power with \(q\equiv 1\pmod{6}\).

This method implements Lemma VII.5.17 from [BJL99] (p.495).

INPUT:

  • q (integer)

  • 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.
  • 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.group_divisible_designs import GDD_4_2
sage: GDD_4_2(7,existence=True)
True
sage: GDD_4_2(7)
Group Divisible Design on 14 points of type 2^7
sage: GDD_4_2(8,existence=True)
Unknown
sage: GDD_4_2(8)
Traceback (most recent call last):
...
NotImplementedError
class sage.combinat.designs.group_divisible_designs.GroupDivisibleDesign(points, groups, blocks, G=None, K=None, lambd=1, check=True, copy=True, **kwds)

Bases: sage.combinat.designs.incidence_structures.IncidenceStructure

Group Divisible Design (GDD)

Let \(K\) and \(G\) be sets of positive integers and let \(\lambda\) be a positive integer. A Group Divisible Design of index \(\lambda\) and order \(v\) is a triple \((V,\mathcal G,\mathcal B)\) where:

  • \(V\) is a set of cardinality \(v\)
  • \(\mathcal G\) is a partition of \(V\) into groups whose size belongs to \(G\)
  • \(\mathcal B\) is a family of subsets of \(V\) whose size belongs to \(K\) such that any two points \(p_1,p_2\in V\) from different groups appear simultaneously in exactly \(\lambda\) elements of \(\mathcal B\). Besides, a group and a block intersect on at most one point.

If \(K=\{k_1,...,k_l\}\) and \(G\) has exactly \(m_i\) groups of cardinality \(k_i\) then \(G\) is said to have type \(k_1^{m_1}...k_l^{m_l}\).

INPUT:

  • points – the underlying set. If points is an integer \(v\), then the set is considered to be \(\{0, ..., v-1\}\).
  • groups – the groups of the design. Set to None for an automatic guess (this triggers check=True and can thus cost some time).
  • blocks – collection of blocks
  • G – list of integers of which the sizes of the groups must be elements. Set to None (automatic guess) by default.
  • 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 indeed a \(GDD\) with the right parameters. Set to True by default.
  • 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: from sage.combinat.designs.group_divisible_designs import GroupDivisibleDesign
sage: TD = designs.transversal_design(4,10)
sage: groups = [list(range(i*10,(i+1)*10)) for i in range(4)]
sage: GDD = GroupDivisibleDesign(40,groups,TD); GDD
Group Divisible Design on 40 points of type 10^4

With unspecified groups:

sage: D = designs.transversal_design(4,3).relabel(list('abcdefghiklm'),inplace=False).blocks()
sage: GDD = GroupDivisibleDesign('abcdefghiklm',None,D)
sage: sorted(GDD.groups())
[['a', 'b', 'c'], ['d', 'e', 'f'], ['g', 'h', 'i'], ['k', 'l', 'm']]
groups()

Return the groups of the Group-Divisible Design.

EXAMPLES:

sage: from sage.combinat.designs.group_divisible_designs import GroupDivisibleDesign
sage: TD = designs.transversal_design(4,10)
sage: groups = [list(range(i*10,(i+1)*10)) for i in range(4)]
sage: GDD = GroupDivisibleDesign(40,groups,TD); GDD
Group Divisible Design on 40 points of type 10^4
sage: GDD.groups()
[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9],
 [10, 11, 12, 13, 14, 15, 16, 17, 18, 19],
 [20, 21, 22, 23, 24, 25, 26, 27, 28, 29],
 [30, 31, 32, 33, 34, 35, 36, 37, 38, 39]]
sage.combinat.designs.group_divisible_designs.group_divisible_design(v, K, G, existence=False, check=False)

Return a \((v,K,G)\)-Group Divisible Design.

A \((v,K,G)\)-GDD is a pair \((\mathcal G, \mathcal B)\) where:

  • \(\mathcal G\) is a partition of \(X=\bigcup \mathcal G\) where \(|X|=v\)
  • \(\forall S\in \mathcal G, |S| \in G\)
  • \(\forall S\in \mathcal B, |S| \in K\)
  • \(\mathcal G\cup \mathcal B\) is a \((v,K\cup G)\)-PBD

For more information, see the documentation of GroupDivisibleDesign or PairwiseBalancedDesign.

INPUT:

  • v (integer)

  • K,G (sets of 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.
  • 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.

Note

The GDD returned by this function are defined on range(v), and its groups are sets of consecutive integers.

EXAMPLES:

sage: designs.group_divisible_design(14,{4},{2})
Group Divisible Design on 14 points of type 2^7