Elementary Properties of Incidence Structures and Designs

Testing the \(t\)-balance: A general Note on Parameter Al

A parameter Al is provided to allow users to specify the algorithm to be used for balance testing. The default value is "NoOrbits", which applies a “brute force” test. One alternative is "Orbits", which uses the orbits of \(t\)-sets under the automorphism group of the incidence structure under consideration. This is much faster than "NoOrbits" for some cases, but slower for others.

The other alternative for checking if an object is \(t\)-balanced is "FastBalanceTest": It is also a “brute force” test but whose implementation allows for a dramatic improvement in efficiency, especially for larger \(t\) (\(t \geq 4\)). This implementation has a drawback however as it may necessitate more memory space than is available, thus resulting in an out-of-memory error. It is for this reason that "FastBalanceTest" is not the default setting for Al, but as a general rule we strongly recommend its usage as we surmise that execution should successfully complete in most cases.

IsSimple(D): Inc -> BoolElt

Returns true if and only if the incidence structure \(D\) has no repeated blocks.

IsTrivial(D): Inc -> BoolElt

Returns true if and only if the incidence structure \(D\) is a trivial incidence structure.

IsSelfDual(D): Inc -> BoolElt

Returns true if and only if the incidence structure \(D\) is self-dual, that is, if \(D\) is isomorphic to its dual.

IsUniform(D): Inc -> BoolElt, RngIntElt

Returns true if and only if the incidence structure \(D\) is uniform; that is, each block contains the same number of points. If true, also returns the blocksize.

IsNearLinearSpace(D): Inc -> BoolElt

Returns true if and only if the incidence structure \(D\) is a near–linear space.

IsLinearSpace(D): Inc -> BoolElt

Returns true if and only if the incidence structure \(D\) is a linear space.

IsDesign(D, t: parameters): Inc, RngIntElt -> BoolElt, RngIntElt
Al: MonStgElt                    Default: "NoOrbits"

Returns true if and only if the incidence structure \(D\) is a \(t\)–design. If true, then the number of blocks of \(D\) containing a general \(t\)-set is also returned. The optional parameter Al can be used to specify the algorithm used for balance testing, see the introduction to Section Elementary Properties of Incidence Structures and Designs for a full description of its usage.

IsBalanced(D, t: parameters): Inc, RngIntElt -> BoolElt, RngIntElt
Al: MonStgElt                    Default: "NoOrbits"

Returns true if and only if the incidence structure \(D\) is balanced (with respect to \(t\)). If true, then the number of blocks of \(D\) containing a general \(t\)-set is also returned. The optional parameter Al can be used to specify the algorithm used for balance testing, see the introduction to Section Elementary Properties of Incidence Structures and Designs for a full description of its usage.

IsComplete(D): Inc -> BoolElt

Return true if and only if \(D\) is the complete design.

IsSymmetric(D): Dsgn -> BoolElt

Return true if and only if the design \(D\) is symmetric.

IsSteiner(D, t): Dsgn, RngIntElt -> BoolElt
Al: MonStg                    Default: "NoOrbits"

Return true if and only if the design \(D\) is a Steiner \(t\)–design. The optional parameter Al can be used to specify the algorithm used for balance testing, see the introduction to Section Elementary Properties of Incidence Structures and Designs for a full description of its usage.

IsPointRegular(D): IncNsp -> BoolElt, RngIntElt

Return true if and only if the (near–)linear space \(D\) is point regular. If true, the point regularity is also returned.

IsLineRegular(D): IncNsp -> BoolElt, RngIntElt

Return true if and only if the (near–)linear space \(D\) is line regular. If true, the line regularity is also returned.