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 (H30E1)
Lattices from Linear Codes
Lattice(C, "A") : Code -> Lat
Lattice(C, "B") : Code -> Lat
Example Lat_Code (H30E2)
Lattices from Algebraic Number Fields
MinkowskiLattice(O) : RngOrd -> Lat, Map
MinkowskiLattice(I) : RngOrdIdl -> Lat, Map
MinkowskiSpace(K) : FldNum -> KModTup, Map
Example Lat_OrderLattice (H30E3)
Special Lattices
Lattice(X, n) : MonStgElt, RngIntElt -> Lat
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 (H30E4)
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
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 (H30E5)
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 (H30E6)
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 (H30E7)
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 (H30E8)
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 (H30E9)
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 (H30E10)
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 (H30E11)
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 (H30E12)
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 (H30E13)
Example Lat_SingularElements (H30E14)
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 (H30E15)
ThetaSeriesIntegral(L, n) : Lat, RngIntElt -> RngSerElt
Lattice Enumeration Utilities
SetVerbose("Enum", v) : MonStgElt, RngIntElt ->
EnumerationCost(L) : Lat -> FldReElt
EnumerationCostArray(L) : Lat -> ModTupFldElt
Example Lat_EnumerationCost (H30E16)
Theta Series as Modular Forms
ThetaSeriesModularFormSpace(L) : Lat -> ModFrm
ThetaSeriesModularForm(L) : Lat -> ModFrmElt
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
VoronoiRelevantVectors(L) : Lat -> [ ModTupFldElt ]
Example Lat_Voronoi (H30E17)
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 (H30E18)
Testing Matrices for Definiteness
IsPositiveDefinite(F) : ModMatRngElt -> BoolElt
IsPositiveSemiDefinite(F) : ModMatRngElt -> BoolElt
IsNegativeDefinite(F) : ModMatRngElt -> BoolElt
IsNegativeSemiDefinite(F) : ModMatRngElt -> BoolElt
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 (H30E19)
Example Lat_Genus (H30E20)
Attributes of Lattices
L`Minimum : Lat -> RngElt
L`MinimumBound : Lat -> RngElt
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 (H30E21)
Accessing the Database
Lattice(D, i: parameters): DB, RngIntElt -> Lattice
LatticeData(D, i): DB, RngIntElt -> Rec
Example Lat_latdb (H30E22)
Hermitian Lattices
HermitianTranspose(M) : Mtrx -> Mtrx
ExpandBasis(M) : Mtrx -> Mtrx
HermitianAutomorphismGroup(M) : Mtrx -> GrpMat
InvariantForms(G) : GrpMat -> SeqEnum
QuaternionicGModule(M, I, J) : ModGrp, AlgMatElt, AlgMatElt -> ModGrp
MooreDeterminant(M) : Mtrx -> Mtrx
Example Lat_coxeter-todd (H30E23)
Example Lat_quaternionic-auto-group (H30E24)
Bibliography
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Version: V2.19 of
Mon Dec 17 14:40:36 EST 2012