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Acknowledgements
Introduction
Presentation of Lattices
Creation of Lattices
Elementary Creation of Lattices
Lattices from Linear Codes
Lattices from Algebraic Number Fields
Special Lattices
Lattice Elements
Creation of Lattice Elements
Operations on Lattice Elements
Predicates and Boolean Operations
Access Operations
Properties of Lattices
Associated Structures
Attributes of Lattices
Predicates and Booleans on Lattices
Base Ring and Base Change
Construction of New Lattices
Sub- and Superlattices and Quotients
Standard Constructions of New Lattices
Reduction of Matrices and Lattices
LLL Reduction
Pair Reduction
Seysen Reduction
HKZ Reduction
Recovering a Short Basis from Short Lattice Vectors
Minima and Element Enumeration
Minimum, Density and Kissing Number
Shortest and Closest Vectors
Short and Close Vectors
Short and Close Vector Processes
Successive Minima and Theta Series
Lattice Enumeration Utilitaries
Voronoi Cells, Holes and Covering Radius
Orthogonalization
Testing Matrices for Definiteness
Automorphism Group and Isometry Testing
Genera and Spinor Genera
Genus Constructions
Invariants of Genera and Spinor Genera
Invariants of p-adic Genera
Neighbour Relations and Graphs
Attributes of Lattices
Lattices from Matrix Groups
Creation of G-Lattices
Operations on G-Lattices
Related Operations on Matrix Groups
Invariant Forms
Endomorphisms
G-invariant Sublattices
Database of Lattices
Creating the Database
Database Information
Accessing the Database
Bibliography
DETAILS
Introduction
Presentation of Lattices
Creation of Lattices
Elementary Creation of Lattices
Lattice(X, M) : ModMatRngElt, AlgMatElt -> Lat
Lattice(X) : ModMatRngElt -> Lat
LatticeWithBasis(B, M) : ModMatRngElt, AlgMatElt -> Lat
LatticeWithBasis(B) : ModMatRngElt -> Lat
LatticeWithGram(F) : AlgMatElt -> Lat
StandardLattice(n) : RngIntElt -> Lat
CoordinateLattice(L) : Lat -> Lat
ScaledLattice(L,n) : Lat, RngIntElt -> Lat
Example Lat_LatticeCreate (H28E1)
Lattices from Linear Codes
Lattice(C, "A") : Code -> Lat
Lattice(C, "B") : Code -> Lat
Example Lat_Code (H28E2)
Lattices from Algebraic Number Fields
MinkowskiLattice(O) : RngOrd -> Lat, Map
MinkowskiLattice(I) : RngOrdIdl -> Lat, Map
MinkowskiSpace(K) : FldNum -> KModTup, Map
Example Lat_OrderLattice (H28E3)
Special Lattices
Lattice(X, n) : MonStgElt, RngIntElt -> Lat
Lattice Elements
Creation of Lattice Elements
L . i : Lat, RngIntElt -> LatElt
L ! Q : Lat, [ RngElt ] -> LatElt
CoordinatesToElement(L, C) : Lat, [ RngIntElt ] -> LatElt
L ! 0 : Lat, RngIntElt -> LatElt
Operations on Lattice Elements
- v : LatElt -> LatElt
v + w : LatElt, LatElt -> LatElt
v - w : LatElt, LatElt -> LatElt
v * s : LatElt, RngIntElt -> .
v / s : LatElt, RngIntElt -> .
v div d : LatElt, RngIntElt -> LatElt
v +:= w : LatElt, LatElt ->
v -:= w : LatElt, LatElt ->
v *:= n : LatElt, RngIntElt ->
v * T : LatElt, AlgMatElt -> LatElt
InnerProduct(v, w) : LatElt, LatElt -> RngElt
Norm(v) : LatElt -> RngElt
Length(v, K) : LatElt, Fld -> FldReElt
Support(v) : LatElt -> SetEnum
Predicates and Boolean Operations
v in L : LatElt, Lat -> BoolElt
v eq w : LatElt, LatElt -> BoolElt
v ne w : LatElt, LatElt -> BoolElt
IsZero(v) : LatElt -> BoolElt
Access Operations
ElementToSequence(v) : LatElt -> [ RngElt ]
Coordinates(v) : LatElt -> [ RngIntElt ]
Coordinates(L, v) : Lat, LatElt -> [ RngIntElt ]
CoordinateVector(v) : LatElt -> LatElt
CoordinateVector(L, v) : Lat, LatElt -> LatElt
Example Lat_LatticeFunctions (H28E4)
Properties of Lattices
Associated Structures
AmbientSpace(L) : Lat -> ModTupFld, Map
CoordinateSpace(L) : Lat -> ModTupFld, Map
Category(L) : Lat -> Cat
Attributes of Lattices
Dimension(L) : Lat -> RngIntElt
Degree(L) : Lat -> RngIntElt
Degree(v) : LatElt -> RngIntElt
Content(L) : Lat -> RngElt
Level(L) : Lat -> RngElt
Determinant(L) : Lat -> RngElt
GramMatrix(L) : Lat -> AlgMatElt
GramMatrix(X) : ModMatRngElt : -> AlgMatElt
InnerProductMatrix(L) : Lat -> AlgMatElt
Basis(L) : Lat -> [ FldReElt ]
BasisMatrix(L) : Lat -> ModMatRngElt
BasisDenominator(L) : Lat -> RngIntElt
QuadraticForm(L) : Lat -> RngMPolElt
Predicates and Booleans on Lattices
L eq M : Lat, Lat -> BoolElt
L ne M : Lat, Lat -> BoolElt
L subset M: Lat, Lat -> BoolElt
IsExact(L) : Lat -> BoolElt
IsIntegral(L) : Lat -> BoolElt
IsEven(L) : Lat -> BoolElt
Base Ring and Base Change
BaseRing(L) : Lat -> Rng
CoordinateRing(L) : Lat -> RngInt
ChangeRing(L, S) : Lat, Rng -> Lat, Map
Construction of New Lattices
Sub- and Superlattices and Quotients
sub<L | S> : Lat, List -> Lat
ext< L | S > : Lat, List -> Lat
T * L : AlgMatElt, Lat -> Lat
s * L : RngElt, Lat -> Lat
L / s : Lat, RngElt -> Lat
quo< L | S > : Lat, List -> GrpAb, Map
L / S : Lat, Lat -> GrpAb, Map
Index(L, S): Lat, Lat -> RngInt
Example Lat_SubSuperQuo (H28E5)
Standard Constructions of New Lattices
Dual(L) : Lat -> Lat
PartialDual(L, n) : Lat, RngIntElt -> Lat
DualBasisLattice(L) : Lat -> Lat
DualQuotient(L) : Lat -> GrpAb, Lat, Map
EvenSublattice(L) : Lat -> Lat, Map
Example Lat_dual (H28E6)
L + M : Lat, Lat -> Lat
L meet M : Lat, Lat -> Lat
DirectSum(L, M) : Lat, Lat -> Lat
OrthogonalDecomposition(L) : Lat -> [Lat]
OrthogonalDecomposition(F) : [Mtrx] -> [* Mtrx *], [* [Mtrx] *]
TensorProduct(L, M) : Lat, Lat -> Lat
ExteriorSquare(L) : Lat -> Lat
SymmetricSquare(L) : Lat -> Lat
PureLattice(L) : Lat -> Lat
IntegralBasisLattice(L) : Lat -> Lat, RngIntElt
Reduction of Matrices and Lattices
LLL Reduction
Example Lat_LLLUsage (H28E7)
LLL(X) : ModMatRngElt -> ModMatRngElt, AlgMatElt, RngIntElt
BasisReduction(X) : ModMatRngElt -> ModMatRngElt, AlgMatElt, RngIntElt
LLLGram(F) : ModMatRngElt -> ModMatRngElt, AlgMatElt, RngIntElt
LLLBasisMatrix(L) : Lat -> ModMatElt, AlgMatElt
LLLGramMatrix(L) : Lat -> AlgMatElt, AlgMatElt
LLL(L) : Lat -> Lat, AlgMatElt
BasisReduction(L) : Lat -> Lat, AlgMatElt
SetVerbose("LLL", v) : MonStgElt, RngIntElt ->
Example Lat_LLLXGCD (H28E8)
Pair Reduction
PairReduce(X) : ModMatRngElt -> ModMatRngElt, AlgMatElt
PairReduceGram(F) : ModMatRngElt -> ModMatRngElt, AlgMatElt, RngIntElt
PairReduce(L) : Lat -> Lat, AlgMatElt
Seysen Reduction
Seysen(X) : ModMatRngElt -> ModMatRngElt, AlgMatElt
SeysenGram(F) : ModMatRngElt -> ModMatRngElt, AlgMatElt, RngIntElt
Seysen(L) : Lat -> Lat, AlgMatElt
Example Lat_Seysen (H28E9)
HKZ Reduction
HKZ(X) : ModMatRngElt -> ModMatRngElt, AlgMatElt
HKZGram(F) : ModMatRngElt -> ModMatRngElt, AlgMatElt
HKZ(L) : Lat -> Lat, AlgMatElt
SetVerbose("HKZ", v) : MonStgElt, RngIntElt ->
GaussReduce(X) : ModMatRngElt -> ModMatRngElt, AlgMatElt
Example Lat_HKZ (H28E10)
Recovering a Short Basis from Short Lattice Vectors
ReconstructLatticeBasis(S, B) : ModMatRngElt, ModMatRngElt -> ModMatRngEltLat
Minima and Element Enumeration
Minimum, Density and Kissing Number
Minimum(L) : Lat -> RngElt
PackingRadius(L) : Lat -> FldReElt
HermiteConstant(n) : RngIntElt -> RngElt
HermiteNumber(L) : Lat -> FldReElt
CentreDensity(L) : Lat -> FldReElt
Density(L) : Lat -> FldReElt
KissingNumber(L) : Lat -> RngElt
Example Lat_Leech (H28E11)
Shortest and Closest Vectors
ShortestVectors(L) : Lat -> [ LatElt ]
ShortestVectorsMatrix(L) : Lat -> ModMatRngElt
ClosestVectors(L, w) : Lat, ModTupRngElt -> [ LatElt ], RngElt
ClosestVectorsMatrix(L, w) : Lat, ModTupRngElt -> ModMatRngElt, RngElt
Example Lat_Closest (H28E12)
Short and Close Vectors
ShortVectors(L, u) : Lat, RngElt -> [ <LatElt, RngElt> ]
ShortVectorsMatrix(L, u) : Lat, RngElt -> ModMatRngElt
CloseVectors(L, w, u) : Lat, ModTupRngElt, RngElt -> [ <LatElt, RngElt> ]
CloseVectorsMatrix(L, w, u) : Lat, ModTupRngElt, RngElt -> ModMatRngElt
Example Lat_Knapsack (H28E13)
Example Lat_SingularElements (H28E14)
Short and Close Vector Processes
ShortVectorsProcess(L, u) : Lat, RngElt -> LatEnumProc
CloseVectorsProcess(L, w, u) : Lat, ModTupRngElt, RngElt -> LatEnumProc
NextVector(P) : LatEnumProc -> LatElt, RngElt
IsEmpty(P) : LatEnumProc -> BoolElt
Successive Minima and Theta Series
SuccessiveMinima(L) : Lat -> [ RngIntElt ], [ LatElt ]
ThetaSeries(L, n) : Lat, RngIntElt -> RngSerElt
Example Lat_ThetaSeries (H28E15)
ThetaSeriesIntegral(L, n) : Lat, RngIntElt -> RngSerElt
Lattice Enumeration Utilitaries
SetVerbose("Enum", v) : MonStgElt, RngIntElt ->
EnumerationCost(L) : Lat -> FldReElt
EnumerationCostArray(L) : Lat -> ModTupFldElt
Example Lat_EnumerationCost (H28E16)
Voronoi Cells, Holes and Covering Radius
VoronoiCell(L) : Lat -> [ ModTupFldElt ], SetEnum , [ ModTupFldElt ]
VoronoiGraph(L) : Lat -> GrphUnd
Holes(L) : Lat -> [ ModTupFldElt ]
DeepHoles(L) : Lat -> [ ModTupFldElt ]
CoveringRadius(L) : Lat -> FldRatElt
Example Lat_Voronoi (H28E17)
Orthogonalization
Orthogonalize(M) : MtrxSpcElt -> MtrxSpcElt, AlgMatElt, RngIntElt
Diagonalization(F) : MtrxSpcElt -> MtrxSpcElt, AlgMatElt, RngIntElt
Orthogonalize(L) : Lat -> Lat, AlgMatElt
Orthonormalize(M, K) : MtrxSpcElt, Fld -> AlgMatElt
Orthonormalize(L, K) : Lat, FldRe -> AlgMatElt
Example Lat_Orthogonalize (H28E18)
Testing Matrices for Definiteness
IsPositiveDefinite(F) : ModMatRngElt -> BoolElt
IsPositiveSemiDefinite(F) : ModMatRngElt -> BoolElt
IsNegativeDefinite(F) : ModMatRngElt -> BoolElt
IsNegativeSemiDefinite(F) : ModMatRngElt -> BoolElt
Automorphism Group and Isometry Testing
AutomorphismGroup(L) : Lat -> GrpMat
AutomorphismGroup(L, F) : Lat, [ AlgMatElt ] -> GrpMat
AutomorphismGroup(F) : [ AlgMatElt ] -> GrpMat
Example Lat_AutoAction (H28E19)
Example Lat_AutoStabilizers (H28E20)
Example Lat_AutoDepth (H28E21)
IsIsometric(L, M) : Lat, Lat -> BoolElt, AlgMatElt
IsIsometric(L, F1, M, F()2) : Lat, [ AlgMatElt ], Lat, [ AlgMatElt ] -> BoolElt, AlgMatElt
IsIsometric(F1, F()2) : [ AlgMatElt ], [ AlgMatElt ] -> BoolElt, AlgMatElt
HasIsometricEmbedding(L, M) : Lat, Lat -> BoolElt, AlgMatElt
Example Lat_Isom (H28E22)
Genera and Spinor Genera
Genus Constructions
Genus(L) : Lat -> SymGen
SpinorGenus(L) : Lat -> SymGen
SpinorGenera(G) : SymGen -> [ SymGen ]
Invariants of Genera and Spinor Genera
Representative(G) : SymGen -> Lat
IsSpinorGenus(G) : SymGen -> BoolElt
IsGenus(G) : SymGen -> BoolElt
Determinant(G) : SymGen -> Lat
LocalGenera(G) : SymGen -> Lat
Representative(G) : SymGen -> Lat
G1 eq G2 : SymGen, SymGen -> BoolElt
# G : SymGen -> RngIntElt
SpinorCharacters(G) : SymGen -> [ GrpDrchElt ]
SpinorGenerators(G) : SymGen -> [ RngIntElt ]
AutomorphousClasses(L,p) : Lat, RngIntElt -> RngIntElt
IsSpinorNorm(G,p) : SymGen, RngIntElt -> RngIntElt
Invariants of p-adic Genera
Prime(G) : SymGenLoc -> RngIntElt
Representative(G) : SymGenLoc -> Lat
Determinant(G) : SymGenLoc -> RngIntElt
Dimension(G) : SymGenLoc -> RngIntElt
G1 eq G2 : SymGenLoc, SymGenLoc -> BoolElt
Neighbour Relations and Graphs
Neighbour(L, v, p) : Lat, LatElt, RngIntElt -> Lat
Neighbours(L, p) : Lat, RngIntElt -> Lat
NeighbourClosure(L, p) : Lat, RngIntElt -> Lat
GenusRepresentatives(L) : Lat -> [ Lat ]
AdjacencyMatrix(G,p) : SymGen, RngIntElt -> AlgMatElt
Example Lat_Neighbour (H28E23)
Example Lat_Genus (H28E24)
Attributes of Lattices
L`Minimum : Lat -> RngElt
L`MinimumBound : Lat -> RngElt
Lattices from Matrix Groups
Creation of G-Lattices
Lattice(G) : GrpMat -> Lat
LatticeWithBasis(G, B) : GrpMat, ModMatRngElt -> Lat
LatticeWithBasis(G, B, M) : GrpMat, ModMatRngElt, AlgMatElt -> Lat
LatticeWithGram(G, F) : GrpMat, AlgMatElt -> Lat
Operations on G-Lattices
IsGLattice(L) : Lat -> GrpMat
Group(L) : Lat -> GrpMat
NumberOfActionGenerators(L) : Lat -> RngIntElt
ActionGenerator(L, i) : Lat, RngIntElt -> GrpMat
NaturalGroup(L) : Lat -> GrpMat
NaturalActionGenerator(L, i) : Lat, RngIntElt -> GrpMat
Related Operations on Matrix Groups
BravaisGroup(G) : GrpMat -> GrpMat
IntegralGroup(G) : GrpMat -> GrpMat, AlgMatElt
Invariant Forms
InvariantForms(G) : GrpMat -> [ AlgMatElt ]
InvariantForms(G, n) : GrpMat, RngIntElt -> [ AlgMatElt ]
SymmetricForms(G) : GrpMat -> [ AlgMatElt ]
SymmetricForms(G, n) : GrpMat, RngIntElt -> [ AlgMatElt ]
AntisymmetricForms(G) : GrpMat -> [ AlgMatElt ]
AntisymmetricForms(G, n) : GrpMat, RngIntElt -> [ AlgMatElt ]
NumberOfInvariantForms(G) : GrpMat -> RngIntElt, RngIntElt
NumberOfSymmetricForms(G) : GrpMat -> RngIntElt
NumberOfAntisymmetricForms(G) : GrpMat -> RngIntElt
PositiveDefiniteForm(G) : GrpMat -> AlgMatElt
Example Lat_PositiveDefiniteForm (H28E25)
Endomorphisms
EndomorphismRing(G) : GrpMat -> AlgMat
Endomorphisms(G, n) : GrpMat, RngIntElt -> [ AlgMatElt ]
DimensionOfEndomorphismRing(G) : GrpMat -> RngIntElt
CentreOfEndomorphismRing(G) : GrpMat -> AlgMat
CentralEndomorphisms(G, n) : GrpMat, RngIntElt -> [ AlgMatElt ]
DimensionOfCentreOfEndomorphismRing(G) : GrpMat -> RngIntElt
G-invariant Sublattices
Sublattices(G, p) : GrpMat, RngIntElt -> [ AlgMatElt ]
Sublattices(G, Q) : GrpMat, [ RngIntElt ] -> [ AlgMatElt ]
Sublattices(G) : GrpMat -> [ AlgMatElt ]
Example Lat_Sublattices (H28E26)
Database of Lattices
Creating the Database
LatticeDatabase() : -> DB
Database Information
# D: DB -> RngIntElt
LargestDimension(D): DB -> RngIntElt
NumberOfLattices(D, d): DB, RngIntElt -> RngIntElt
NumberOfLattices(D, N): DB, MonStgElt -> RngIntElt
LatticeName(D, i): DB, RngIntElt -> MonStgElt, RngIntElt
LatticeName(D, d, i): DB, RngIntElt, RngIntElt -> RecMonStgElt, RngIntElt
LatticeName(D, N): DB, MonStgElt -> RecMonStgElt, RngIntElt
LatticeName(D, N, i): DB, MonStgElt, RngIntElt -> RecMonStgElt, RngIntElt
Example Lat_latdb-names (H28E27)
Accessing the Database
Lattice(D, i: parameters): DB, RngIntElt -> Lattice
LatticeData(D, i): DB, RngIntElt -> Rec
Bibliography
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