Elementary Properties of Incidence Structures and Designs

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. The default value is "NoOrbits", which applies a "brute force" test. The alternative is "Orbits", which uses the orbits of t--sets under the automorphism group of D. This is much faster for some cases, but slower for others.

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 may be used to specify the algorithm used. The default value is "NoOrbits", which applies a "brute force" test. The alternative is "Orbits", which uses the orbits of t--sets under the automorphism group of D. This is much faster for some cases, but slower for others.

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 -> BoolElt

Return true if and only if the design D is a Steiner t--design.

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.