Covering designs: coverings of \(t\)-element subsets of a \(v\)-set by \(k\)-sets

A \((v, k, t)\) covering design \(C\) is an incidence structure consisting of a set of points \(P\) of order \(v\), and a set of blocks \(B\), where each block contains \(k\) points of \(P\). Every \(t\)-element subset of \(P\) must be contained in at least one block.

If every \(t\)-set is contained in exactly one block of \(C\), then we have a block design. Following the block design implementation, the standard representation of a covering design uses \(P = [0, 1, ..., v-1]\).

In addition to the parameters and incidence structure for a covering design from this database, we include extra information:

  • Best known lower bound on the size of a \((v, k, t)\)-covering design
  • Name of the person(s) who produced the design
  • Method of construction used
  • Date when the design was added to the database

REFERENCES:

[1]La Jolla Covering Repository, https://ljcr.dmgordon.org/cover.html
[2]Daniel M. Gordon and Douglas R. Stinson, Coverings, Chapter 1 in: Charles J. Colbourn and Jeffrey H. Dinitz, Handbook of Combinatorial Designs, 2nd Edition, 2006. https://www.dmgordon.org/papers/hcd.pdf

AUTHORS:

  • Daniel M. Gordon (2008-12-22): initial version

Classes and methods

class sage.combinat.designs.covering_design.CoveringDesign(v=0, k=0, t=0, size=0, points=[], blocks=[], low_bd=0, method='', creator='', timestamp='')

Bases: sage.structure.sage_object.SageObject

Covering design.

INPUT:

  • v, k, t – integer parameters of the covering design
  • size (integer)
  • points – list of points (default points are \([0, ..., v-1]\))
  • blocks
  • low_bd (integer) – lower bound for such a design
  • method, creator, timestamp – database information
creator()

Return the creator of the covering design

This field is optional, and is used in a database to give attribution for the covering design It can refer to the person who submitted it, or who originally gave a construction

EXAMPLES:

sage: from sage.combinat.designs.covering_design import CoveringDesign
sage: C = CoveringDesign(7, 3, 2, 7, range(7), [[0, 1, 2],
....:     [0, 3, 4], [0, 5, 6], [1, 3, 5], [1, 4, 6], [2, 3, 6],
....:     [2, 4, 5]],0, 'Projective Plane', 'Gino Fano')
sage: C.creator()
'Gino Fano'
incidence_structure()

Return the incidence structure of this design, without extra parameters.

EXAMPLES:

sage: from sage.combinat.designs.covering_design import CoveringDesign
sage: C = CoveringDesign(7, 3, 2, 7, range(7), [[0, 1, 2],
....:     [0, 3, 4], [0, 5, 6], [1, 3, 5], [1, 4, 6],
....:     [2, 3, 6], [2, 4, 5]], 0, 'Projective Plane')
sage: D = C.incidence_structure()
sage: D.ground_set()
[0, 1, 2, 3, 4, 5, 6]
sage: D.blocks()
[[0, 1, 2], [0, 3, 4], [0, 5, 6], [1, 3, 5],
[1, 4, 6], [2, 3, 6], [2, 4, 5]]
is_covering()

Check all \(t\)-sets are in fact covered by the blocks of self.

Note

This is very slow and wasteful of memory.

EXAMPLES:

sage: C = CoveringDesign(7, 3, 2, 7, range(7), [[0, 1, 2],
....:     [0, 3, 4], [0, 5, 6], [1, 3, 5], [1, 4, 6],
....:     [2, 3, 6], [2, 4, 5]], 0, 'Projective Plane')
sage: C.is_covering()
True
sage: C = CoveringDesign(7, 3, 2, 7, range(7), [[0, 1, 2],
....:     [0, 3, 4], [0, 5, 6], [1, 3, 5], [1, 4, 6], [2, 3, 6],
....:     [2, 4, 6]], 0, 'not a covering')   # last block altered
sage: C.is_covering()
False
k()

Return \(k\), the size of blocks of the covering design

EXAMPLES:

sage: from sage.combinat.designs.covering_design import CoveringDesign
sage: C = CoveringDesign(7, 3, 2, 7, range(7), [[0, 1, 2],
....:     [0, 3, 4], [0, 5, 6], [1, 3, 5], [1, 4, 6],
....:     [2, 3, 6], [2, 4, 5]], 0, 'Projective Plane')
sage: C.k()
3
low_bd()

Return a lower bound for the number of blocks a covering design with these parameters could have.

Typically this is the Schonheim bound, but for some parameters better bounds have been shown.

EXAMPLES:

sage: from sage.combinat.designs.covering_design import CoveringDesign
sage: C = CoveringDesign(7, 3, 2, 7, range(7), [[0, 1, 2],
....:     [0, 3, 4], [0, 5, 6], [1, 3, 5], [1, 4, 6],
....:     [2, 3, 6], [2, 4, 5]], 0, 'Projective Plane')
sage: C.low_bd()
7
method()

Return the method used to create the covering design.

This field is optional, and is used in a database to give information about how coverings were constructed.

EXAMPLES:

sage: from sage.combinat.designs.covering_design import CoveringDesign
sage: C = CoveringDesign(7, 3, 2, 7, range(7), [[0, 1, 2],
....:     [0, 3, 4], [0, 5, 6], [1, 3, 5], [1, 4, 6],
....:     [2, 3, 6], [2, 4, 5]], 0, 'Projective Plane')
sage: C.method()
'Projective Plane'
size()

Return the number of blocks in the covering design

EXAMPLES:

sage: from sage.combinat.designs.covering_design import CoveringDesign
sage: C = CoveringDesign(7, 3, 2, 7, range(7), [[0, 1, 2],
....:     [0, 3, 4], [0, 5, 6], [1, 3, 5], [1, 4, 6],
....:     [2, 3, 6], [2, 4, 5]], 0, 'Projective Plane')
sage: C.size()
7
t()

Return \(t\), the size of sets which must be covered by the blocks of the covering design

EXAMPLES:

sage: from sage.combinat.designs.covering_design import CoveringDesign
sage: C = CoveringDesign(7, 3, 2, 7, range(7), [[0, 1, 2],
....:     [0, 3, 4], [0, 5, 6], [1, 3, 5], [1, 4, 6],
....:     [2, 3, 6], [2, 4, 5]], 0, 'Projective Plane')
sage: C.t()
2
timestamp()

Return the time that the covering was submitted to the database

EXAMPLES:

sage: from sage.combinat.designs.covering_design import CoveringDesign
sage: C = CoveringDesign(7, 3, 2, 7, range(7), [[0, 1, 2],
....:     [0, 3, 4], [0, 5, 6], [1, 3, 5], [1, 4, 6],
....:     [2, 3, 6], [2, 4, 5]],0, 'Projective Plane',
....:     'Gino Fano', '1892-01-01 00:00:00')
sage: C.timestamp()  # No exact date known; in Fano's 1892 article
'1892-01-01 00:00:00'
v()

Return \(v\), the number of points in the covering design.

EXAMPLES:

sage: from sage.combinat.designs.covering_design import CoveringDesign
sage: C = CoveringDesign(7, 3, 2, 7, range(7), [[0, 1, 2],
....:     [0, 3, 4], [0, 5, 6], [1, 3, 5], [1, 4, 6],
....:     [2, 3, 6], [2, 4, 5]], 0, 'Projective Plane')
sage: C.v()
7
sage.combinat.designs.covering_design.best_known_covering_design_www(v, k, t, verbose=False)

Return the best known \((v, k, t)\) covering design from an online database.

This uses the La Jolla Covering Repository, a database available at https://ljcr.dmgordon.org/cover.html

INPUT:

  • v – integer, the size of the point set for the design
  • k – integer, the number of points per block
  • t – integer, the size of sets covered by the blocks
  • verbose – bool (default: False), print verbose message

OUTPUT:

A CoveringDesign object representing the (v, k, t)-covering design with smallest number of blocks available in the database.

EXAMPLES:

sage: from sage.combinat.designs.covering_design import (  # optional - internet
....:     best_known_covering_design_www)
sage: C = best_known_covering_design_www(7, 3, 2)  # optional - internet
sage: print(C)                                     # optional - internet
C(7, 3, 2) = 7
Method: lex covering
Submitted on: 1996-12-01 00:00:00
0  1  2
0  3  4
0  5  6
1  3  5
1  4  6
2  3  6
2  4  5

A ValueError is raised if the (v, k, t) parameters are not found in the database.

sage.combinat.designs.covering_design.schonheim(v, k, t)

Return the Schonheim lower bound for the size of such a covering design.

INPUT:

  • v – integer, size of point set
  • k – integer, cardinality of each block
  • t – integer, cardinality of sets being covered

OUTPUT:

The Schonheim lower bound for such a covering design’s size: \(C(v, k, t) \leq \lceil(\frac{v}{k} \lceil \frac{v-1}{k-1} \cdots \lceil \frac{v-t+1}{k-t+1} \rceil \cdots \rceil \rceil\)

EXAMPLES:

sage: from sage.combinat.designs.covering_design import schonheim
sage: schonheim(10, 3, 2)
17
sage: schonheim(32, 16, 8)
930
sage.combinat.designs.covering_design.trivial_covering_design(v, k, t)

Construct a trivial covering design.

INPUT:

  • v – integer, size of point set
  • k – integer, cardinality of each block
  • t – integer, cardinality of sets being covered

OUTPUT:

A trivial \((v, k, t)\) covering design

EXAMPLES:

sage: C = trivial_covering_design(8, 3, 1)
sage: print(C)
C(8, 3, 1) = 3
Method: Trivial
0   1   2
0   6   7
3   4   5
sage: C = trivial_covering_design(5, 3, 2)
sage: print(C)
4 <= C(5, 3, 2) <= 10
Method: Trivial
0   1   2
0   1   3
0   1   4
0   2   3
0   2   4
0   3   4
1   2   3
1   2   4
1   3   4
2   3   4

NOTES:

Cases are:

  • \(t=0\): This could be empty, but it’s a useful convention to have one block (which is empty if $k=0$).
  • \(t=1\) : This contains \(\lceil v/k \rceil\) blocks: \([0, ..., k-1], [k, ..., 2k-1], ...\). The last block wraps around if \(k\) does not divide \(v\).
  • anything else: Just use every \(k\)-subset of \([0, 1,..., v-1]\).