[Next][Prev] [Right] [Left] [Up] [Index] [Root]

INCIDENCE STRUCTURES AND DESIGNS

 
Acknowledgements
 
Introduction
 
Construction of Incidence Structures and Designs
 
The Point-Set and Block-Set of an Incidence Structure
      Introduction
      Creating Point-Sets and Block-Sets
      Creating Points and Blocks
 
General Design Constructions
      The Construction of Related Structures
      The Witt Designs
      Difference Sets and their Development
 
Elementary Invariants of an Incidence Structure
 
Elementary Invariants of a Design
 
Operations on Points and Blocks
 
Elementary Properties of Incidence Structures and Designs
 
Resolutions, Parallelisms and Parallel Classes
 
Conversion Functions
 
Identity and Isomorphism
 
The Automorphism Group of an Incidence Structure
      Construction of Automorphism Groups
      Action of Automorphisms
 
Incidence Structures, Graphs and Codes
 
Automorphisms of Matrices
 
Bibliography







DETAILS

 
Introduction

 
Construction of Incidence Structures and Designs
      IncidenceStructure< v | X > : RngIntElt, List -> Inc
      NearLinearSpace< v | X : parameters > : RngIntElt, List -> IncNsp
      LinearSpace< v | X : parameters > : RngIntElt, List -> IncLsp
      Design< t, v | X : parameters > : RngIntElt, RngIntElt, List -> Dsgn
      Example Design_Constructors (H147E1)

 
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
            Random(P) : IncPtSet -> IncPt
            P ! x : IncPtSet, Elt -> Incpt
            Block(D, i) : Inc, RngIntElt -> IncBlk
            B . i : IncBlkSet, RngIntElt -> IncBlk
            Representative(B) : IncBlkSet -> IncBlk
            Random(B) : IncBlkSet -> IncBlk
            B ! S : IncBlkSet, SetEnum -> IncBlk
            Representative(b) : IncBlk -> IncPt
            Random(b) : IncBlk -> IncPt
            Example Design_points-blocks (H147E2)

 
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 Design_related (H147E3)

      The Witt Designs
            WittDesign(n) : RngIntElt -> Dsgn
            Example Design_wittex (H147E4)

      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 Design_DevelopDifferenceSet (H147E5)

 
Elementary Invariants of an Incidence Structure
      NumberOfPoints(D) : Inc -> RngInt
      Points(D) : Inc -> { IncPt }
      Support(D) : Inc -> { Elt }
      PointDegrees(D) : Inc -> [ RngIntElt ]
      NumberOfBlocks(D) : Inc -> RngIntElt
      Blocks(D) : Inc -> { IncBlk }
      BlockDegrees(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
      Covalence(D, s) : Dsgn, RngIntElt -> RngIntElt
      Order(D) : Dsgn -> RngIntElt
      IntersectionNumber(D, i, j) : Dsgn, RngIntElt, RngIntElt -> RngIntElt
      PascalTriangle(D) : Dsgn -> SeqEnum
      Example Design_design-invar (H147E6)

 
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
      Set(B) : IncBlk -> { IncPt }
      Support(B) : IncBlk -> { Elt }
      IsBlock(D, S) : Inc, IncBlk -> BoolElt, IncBlk
      Line(D, p, q) : Inc, IncPt, IncPt -> IncBlk
      ConnectionNumber(D, p, B) : Inc, IncPt, IncBlk -> RngIntElt
      Example Design_pts-blks-ops (H147E7)

 
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 -> 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 Design_resol-parallel (H147E8)

 
Conversion Functions
      IncidenceStructure(I) : Inc -> Inc
      NearLinearSpace(I) : Inc -> IncNsp
      LinearSpace(I) : Inc -> IncLsp
      Design(I, t) : Inc, RngIntElt -> Dsgn
      Example Design_conv (H147E9)

 
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 Design_auto (H147E10)

      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, 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 Design_automorphism (H147E11)

 
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 Design_graphs (H147E12)

 
Automorphisms of Matrices
      M ^ x : Mtrx, GrpPermElt -> Mtrx
      AutomorphismGroup(M) : Mtrx -> GrpPerm
      IsIsomorphic(M, N) : Mtrx, Mtrx -> BoolElt, GrpPermElt
      Example Design_FanoAuto (H147E13)

 
Bibliography

[Next][Prev] [Right] [____] [Up] [Index] [Root]
Version: V2.19 of Wed Apr 24 15:09:57 EST 2013