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
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)
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
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Version: V2.19 of
Mon Dec 17 14:40:36 EST 2012