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=None, blocks=None, low_bd=0, method='', creator='', timestamp='')¶
Bases:
sage.structure.sage_object.SageObject
Covering design.
INPUT:
v
,k
,t
– integer parameters of the covering designsize
(integer)points
– list of points (default points are \([0, ..., v-1]\))blocks
low_bd
(integer) – lower bound for such a designmethod
,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 designk
– integer, the number of points per blockt
– integer, the size of sets covered by the blocksverbose
– 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 setk
– integer, cardinality of each blockt
– 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 setk
– integer, cardinality of each blockt
– 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
Note
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 \([0, 1,..., v-1]\).