Incidence Structures and Designs
- Introduction
- Construction of Incidence Structures and Designs
IncidenceStructure< v | X >: RngIntElt, List → IncIncidenceStructure< P | X >: SetIndx, List → IncNearLinearSpace< v | X : parameters >: RngIntElt, List → IncNspNearLinearSpace< P | X : parameters >: SetIndx, List → IncNspLinearSpace< v | X : parameters >: RngIntElt, List → IncLspLinearSpace< P | X : parameters >: SetIndx, List → IncLspDesign< t, v | X : parameters >: RngIntElt, RngIntElt, List → DsgnDesign< t, P | X : parameters >: RngIntElt, SetIndx, List → Dsgn- Example: Constructing the Fano Plane
- The Point-Set and Block-Set of an Incidence Structure
- General Design Constructions
- The Construction of Related Structures
Complement(D): Inc → IncDual(D): Inc → IncContraction(D, p): Inc, IncPt → IncContraction(D, b): Inc, IncBlk → IncResidual(D, b): Inc, IncBlk → IncResidual(D, p): Inc, IncPt → IncSimplify(D): Inc → IncSum(Q): [ Inc ] → IncUnion(D, E): Inc, Inc → IncRestriction(D, S): IncNsp, { Incpt } → IncNsp- Example: Related Structures of a Design
- The Witt Designs
- Difference Sets and their Development
- Elementary Invariants of an Incidence Structure
- Elementary Invariants of a Design
- Operations on Points and Blocks
p in B: IncPt, IncBlk → BoolEltp notin B: IncPt, IncBlk → BoolEltS subset B: { IncPt }, IncBlk → BoolEltS notsubset B: { IncPt }, IncBlk → BoolEltPointDegree(D, p): Inc, IncPt → RngIntEltBlockDegree(D, B): Inc, IncBlk → RngIntEltBlockSize(D, B): Inc, IncBlk → RngIntElt# B: IncBlk → RngIntEltSet(B): IncBlk → { IncPt }Support(B): IncBlk → { Elt }IsBlock(D, S): Inc, IncBlk → BoolElt, IncBlkIsBlock(D, S): Inc, SetEnum → BoolElt, IncBlkLine(D, p, q): Inc, IncPt, IncPt → IncBlkBlock(D, p, q): Inc, IncPt, IncPt → IncBlkConnectionNumber(D, p, B): Inc, IncPt, IncBlk → RngIntElt- Example: Operations on Points and Blocks
- Elementary Properties of Incidence Structures and Designs
IsSimple(D): Inc → BoolEltIsTrivial(D): Inc → BoolEltIsSelfDual(D): Inc → BoolEltIsUniform(D): Inc → BoolElt, RngIntEltIsNearLinearSpace(D): Inc → BoolEltIsLinearSpace(D): Inc → BoolEltIsDesign(D, t: parameters): Inc, RngIntElt → BoolElt, RngIntEltIsBalanced(D, t: parameters): Inc, RngIntElt → BoolElt, RngIntEltIsComplete(D): Inc → BoolEltIsSymmetric(D): Dsgn → BoolEltIsSteiner(D, t): Dsgn, RngIntElt → BoolEltIsPointRegular(D): IncNsp → BoolElt, RngIntEltIsLineRegular(D): IncNsp → BoolElt, RngIntElt
- Resolutions, Parallelisms, and Parallel Classes
HasResolution(D): Inc → BoolElt, { SetEnum}, RngIntEltHasResolution(D, λ): Inc, RngIntElt → BoolElt, { SetEnum}AllResolutions(D): Inc → SeqEnumAllResolutions(D, λ): Inc, RngIntElt → SeqEnumIsResolution(D, P): Inc, SetEnum[SetEnum] → BoolElt, RngIntEltHasParallelism(D: parameters): Inc, RngIntElt → BoolElt, { SetEnum}AllParallelisms(D): Inc → SeqEnumIsParallelism(D, P): Inc, SetEnum[SetEnum] → BoolElt, RngIntEltHasParallelClass(D): Inc → BoolElt, { IncBlk}IsParallelClass(D, B, C): Inc, IncBlk, IncBlk → BoolElt, { IncBlk}AllParallelClasses(D): Inc → SeqEnum- Example: Resolutions and Parallel Classes
- Conversion Functions
- Identity and Isomorphism
- The Automorphism Group of an Incidence Structure
- Construction of Automorphism Groups
AutomorphismGroup(D): Inc → GrpPerm, GSet, GSet, PowMap, MapAutomorphismSubgroup(D): Inc → GrpPerm, PowMap, MapAutomorphismGroupStabilizer(D, k): Inc, RngIntElt → GrpPerm, PowMap, MapPointGroup(D): Inc → GrpPerm, GSetBlockGroup(D): Inc → GrpPermAut(D): Inc → PowMapAut, Map- Example: Automorphism Group of a Hadamard Design
- Action of Automorphisms
Image(g, Y, y): GrpPermElt, GSet, Elt → EltOrbit(G, Y, y): GrpPerm, GSet, Elt → GSetOrbits(G, Y): GrpPerm, GSet → [ GSet ]Stabilizer(G, Y, y): GrpPerm, GSet, Elt → GrpPermAction(G, Y): GrpPerm, GSet → Hom(Grp), GrpPerm, GrpPermActionImage(G, Y): GrpPerm, GSet → GrpPermActionKernel(G, Y): GrpPerm, GSet → GrpPermIsPointTransitive(D): Inc → BoolEltIsBlockTransitive(D): Inc → BoolElt- Example: Automorphism Group of a Witt Design
- Incidence Structures, Graphs, and Codes
- Automorphisms of Matrices
- Bibliography