Incidence Structures and Designs

• Introduction
• Construction of Incidence Structures and Designs
  • IncidenceStructure< v | X >: RngIntElt, List → Inc
  • IncidenceStructure< P | X >: SetIndx, List → Inc
  • NearLinearSpace< v | X : parameters >: RngIntElt, List → IncNsp
  • NearLinearSpace< P | X : parameters >: SetIndx, List → IncNsp
  • LinearSpace< v | X : parameters >: RngIntElt, List → IncLsp
  • LinearSpace< P | X : parameters >: SetIndx, List → IncLsp
  • Design< t, v | X : parameters >: RngIntElt, RngIntElt, List → Dsgn
  • Design< t, P | X : parameters >: RngIntElt, SetIndx, List → Dsgn
  • Example: Constructors
• The Point-Set and Block-Set of an Incidence Structure
  • Introduction
  • Creating Point-Sets and Block-Sets
    • PointSet(D): Inc → IncPtSet
    • BlockSet(D): Inc → IncBlkSet
  • Creating Points and Blocks
    • Point(D, i): Inc, RngIntElt → IncPt
    • P . i: IncPtSet, RngIntElt → IncPt
    • Representative(P): IncPtSet → IncPt
    • Rep(P): IncPtSet → IncPt
    • Random(P): IncPtSet → IncPt
    • P ! x: IncPtSet, Elt → Incpt
    • Block(D, i): Inc, RngIntElt → IncBlk
    • B . i: IncBlkSet, RngIntElt → IncBlk
    • Representative(B): IncBlkSet → IncBlk
    • Rep(B): IncBlkSet → IncBlk
    • Random(B): IncBlkSet → IncBlk
    • B ! S: IncBlkSet, SetEnum → IncBlk
    • Representative(b): IncBlk → IncPt
    • Rep(b): IncBlk → IncPt
    • Random(b): IncBlk → IncPt
    • Example: Points Blocks
• General Design Constructions
  • The Construction of Related Structures
    • Complement(D): Inc → Inc
    • Dual(D): Inc → Inc
    • Contraction(D, p): Inc, IncPt → Inc
    • Contraction(D, b): Inc, IncBlk → Inc
    • Residual(D, b): Inc, IncBlk → Inc
    • Residual(D, p): Inc, IncPt → Inc
    • Simplify(D): Inc → Inc
    • Sum(Q): [ Inc ] → Inc
    • Union(D, E): Inc, Inc → Inc
    • Restriction(D, S): IncNsp, { Incpt } → IncNsp
    • Example: related
  • The Witt Designs
    • WittDesign(n): RngIntElt → Dsgn
    • Example: wittex
  • Difference Sets and their Development
    • DifferenceSet(p, t): RngIntElt, MonStgElt → { RngIntResElt }
    • SingerDifferenceSet(n, q): RngIntElt, RngIntElt → { RngIntResElt }
    • IsDifferenceSet(B): SetEnum → BoolElt, RngIntElt
    • Development(B): {  RngElt } → Inc
    • Development(T): { { Elt } } → Inc
    • Example: Develop Difference Set
• Elementary Invariants of an Incidence Structure
  • NumberOfPoints(D): Inc → RngInt
  • # P: IncPtSet → RngIntElt
  • Points(D): Inc → { IncPt }
  • Support(D): Inc → { Elt }
  • PointDegrees(D): Inc → [ RngIntElt ]
  • NumberOfBlocks(D): Inc → RngIntElt
  • # B: IncBlkSet → RngIntElt
  • Blocks(D): Inc → { IncBlk}
  • BlockDegrees(D): Inc → [ RngIntElt ]
  • BlockSizes(D): Inc → [ RngIntElt ]
  • Covalence(D, S): Inc, { IncPt } → RngIntElt
  • IncidenceMatrix(D): Inc → ModMatRngElt
  • pRank(D, p): Inc, RngIntElt → RngIntElt
• Elementary Invariants of a Design
  • Parameters(D): Dsgn → Record
  • ReplicationNumber(D): Dsgn → RngIntElt
  • BlockDegree(D): Dsgn → RngIntElt
  • BlockSize(D): Dsgn → RngIntElt
  • Covalence(D, s): Dsgn, RngIntElt → RngIntElt
  • Order(D): Dsgn → RngIntElt
  • IntersectionNumber(D, i, j): Dsgn, RngIntElt, RngIntElt → RngIntElt
  • PascalTriangle(D): Dsgn → SeqEnum
  • Example: Design Invar
• Operations on Points and Blocks
  • p in B: IncPt, IncBlk → BoolElt
  • p notin B: IncPt, IncBlk → BoolElt
  • S subset B: { IncPt }, IncBlk → BoolElt
  • S notsubset B: { IncPt }, IncBlk → BoolElt
  • PointDegree(D, p): Inc, IncPt → RngIntElt
  • BlockDegree(D, B): Inc, IncBlk → RngIntElt
  • BlockSize(D, B): Inc, IncBlk → RngIntElt
  • # B: IncBlk → RngIntElt
  • Set(B): IncBlk → { IncPt }
  • Support(B): IncBlk → { Elt }
  • IsBlock(D, S): Inc, IncBlk → BoolElt, IncBlk
  • IsBlock(D, S): Inc, SetEnum → BoolElt, IncBlk
  • Line(D, p, q): Inc, IncPt, IncPt → IncBlk
  • Block(D, p, q): Inc, IncPt, IncPt → IncBlk
  • ConnectionNumber(D, p, B): Inc, IncPt, IncBlk → RngIntElt
  • Example: Pts Blks Ops
• Elementary Properties of Incidence Structures and Designs
  • IsSimple(D): Inc → BoolElt
  • IsTrivial(D): Inc → BoolElt
  • IsSelfDual(D): Inc → BoolElt
  • IsUniform(D): Inc → BoolElt, RngIntElt
  • IsNearLinearSpace(D): Inc → BoolElt
  • IsLinearSpace(D): Inc → BoolElt
  • IsDesign(D, t: parameters): Inc, RngIntElt → BoolElt, RngIntElt
  • IsBalanced(D, t: parameters): Inc, RngIntElt → BoolElt, RngIntElt
  • IsComplete(D): Inc → BoolElt
  • IsSymmetric(D): Dsgn → BoolElt
  • IsSteiner(D, t): Dsgn, RngIntElt → BoolElt
  • IsPointRegular(D): IncNsp → BoolElt, RngIntElt
  • IsLineRegular(D): IncNsp → BoolElt, RngIntElt
• Resolutions, Parallelisms and Parallel Classes
  • HasResolution(D): Inc → BoolElt, { SetEnum}, RngIntElt
  • HasResolution(D, λ): Inc, RngIntElt → BoolElt, { SetEnum}
  • AllResolutions(D): Inc → SeqEnum
  • AllResolutions(D, λ): Inc, RngIntElt → SeqEnum
  • IsResolution(D, P): Inc, SetEnum[SetEnum] → BoolElt, RngIntElt
  • HasParallelism(D: parameters): Inc, RngIntElt → BoolElt, { SetEnum}
  • AllParallelisms(D): Inc → SeqEnum
  • IsParallelism(D, P): Inc, SetEnum[SetEnum] → BoolElt, RngIntElt
  • HasParallelClass(D): Inc → BoolElt, { IncBlk}
  • IsParallelClass(D, B, C): Inc, IncBlk, IncBlk → BoolElt, { IncBlk}
  • AllParallelClasses(D): Inc → SeqEnum
  • Example: Resol Parallel
• Conversion Functions
  • IncidenceStructure(I): Inc → Inc
  • NearLinearSpace(I): Inc → IncNsp
  • LinearSpace(I): Inc → IncLsp
  • Design(I, t): Inc, RngIntElt → Dsgn
  • Example: conv
• Identity and Isomorphism
  • D eq E: Inc, Inc → BoolElt
  • D ne E: Inc, Inc → BoolElt
  • IsIsomorphic(D, E: parameters): Inc, Inc → BoolElt, Map
• The Automorphism Group of an Incidence Structure
  • Construction of Automorphism Groups
    • AutomorphismGroup(D): Inc → GrpPerm, GSet, GSet, PowMap, Map
    • AutomorphismSubgroup(D): Inc → GrpPerm, PowMap, Map
    • AutomorphismGroupStabilizer(D, k): Inc, RngIntElt → GrpPerm, PowMap, Map
    • PointGroup(D): Inc → GrpPerm, GSet
    • BlockGroup(D): Inc → GrpPerm
    • Aut(D): Inc → PowMapAut, Map
    • Example: auto
  • Action of Automorphisms
    • Image(g, Y, y): GrpPermElt, GSet, Elt → Elt
    • Orbit(G, Y, y): GrpPerm, GSet, Elt → GSet
    • Orbits(G, Y): GrpPerm, GSet → [ GSet ]
    • Stabilizer(G, Y, y): GrpPerm, GSet, Elt → GrpPerm
    • Action(G, Y): GrpPerm, GSet → Hom(Grp), GrpPerm, GrpPerm
    • ActionImage(G, Y): GrpPerm, GSet → GrpPerm
    • ActionKernel(G, Y): GrpPerm, GSet → GrpPerm
    • IsPointTransitive(D): Inc → BoolElt
    • IsBlockTransitive(D): Inc → BoolElt
    • Example: automorphism
• Incidence Structures, Graphs and Codes
  • IncidenceStructure(G): Grph → Inc
  • PointGraph(D): Inc → Grph
  • BlockGraph(D): Inc → Grph
  • IncidenceGraph(D): Inc → Grph
  • LinearCode(D, K): Inc, FldFin → Code
  • Example: graphs
• Automorphisms of Matrices
  • M ^ x: Mtrx, GrpPermElt → Mtrx
  • AutomorphismGroup(M : parameters): Mtrx → GrpPerm
  • IsIsomorphic(M, N: parameters): Mtrx, Mtrx → BoolElt, GrpPermElt
  • Example: Fano Auto