Lattices
- Introduction
- Presentation of Lattices
- Creation of Lattices
- Elementary Creation of Lattices
Lattice(X, M): ModMatRngElt, AlgMatElt → LatLattice(n, Q, M): RngIntElt, [ RngElt ], AlgMatElt → LatLattice(S, M): ModTupRng, AlgMatElt → LatLattice(X): ModMatRngElt → LatLattice(n, Q): RngIntElt, [ RngElt ] → LatLattice(S): ModTupRng → LatLatticeWithBasis(B, M): ModMatRngElt, AlgMatElt → LatLatticeWithBasis(n, Q, M): RngIntElt, [ RngElt ], AlgMatElt → LatLatticeWithBasis(S, M): ModTupRng, AlgMatElt → LatLatticeWithBasis(B): ModMatRngElt → LatLatticeWithBasis(n, Q): RngIntElt, [ RngElt ] → LatLatticeWithBasis(S): ModTupRng → LatLatticeWithGram(F): AlgMatElt → LatLatticeWithGram(n, Q): RngIntElt, [RngElt] → LatStandardLattice(n): RngIntElt → LatCoordinateLattice(L): Lat → LatModule(L): Lat → LatZLattice(L): Lat → LatScaledLattice(L, n): Lat, RngIntElt → LatScaledLattice(L, n): Lat, FldRatElt → LatTernaryQuadraticLattice(d): RngIntElt → LatQuaternaryLatticeOfPrimeDiscriminant(p): RngIntElt → LatQuinaryLatticeOfPrimeDiscriminant(p): RngIntElt → LatQuadraticLattice(n, d): RngIntElt, RngIntElt → LatRandomLattice(n, M): RngIntElt, RngIntElt → LatExample: Lattice Create
- Lattices from Linear Codes
- Lattices from Algebraic Number Fields
- Special Lattices
- Lattice Elements
- Creation of Lattice Elements
L . i: Lat, RngIntElt → LatEltL ! Q: Lat, [ RngElt ] → LatEltelt<L | Q>: Lat, [ RngElt ] → LatEltCoordinatesToElement(L, C): Lat, [ RngIntElt ] → LatEltCoordelt(L, C): Lat, [ RngIntElt ] → LatEltCoordinatesToElement(L, C): Lat, Mtrx → LatEltCoordelt(L, C): Lat, Mtrx → LatEltL ! 0: Lat, RngIntElt → LatEltZero(L): Lat → LatElt
- Operations on Lattice Elements
- v: LatElt → LatEltv + w: LatElt, LatElt → LatEltv - w: LatElt, LatElt → LatEltv * s: LatElt, RngIntElt → .s * v: RngIntElt, LatElt → .v / s: LatElt, RngIntElt → .v div d: LatElt, RngIntElt → LatEltv +:= w: LatElt, LatEltv -:= w: LatElt, LatEltv *:= n: LatElt, RngIntEltv * T: LatElt, AlgMatElt → LatEltv * T: LatElt, ModMatRngElt → LatEltv * T: LatElt, GrpMatElt → LatEltInnerProduct(v, w): LatElt, LatElt → RngElt(v, w)Norm(v): LatElt → RngEltLength(v, K): LatElt, Fld → FldReEltLength(v): LatElt → FldReEltSupport(v): LatElt → SetEnum
- Predicates and Boolean Operations
- Access Operations
- Properties of Lattices
- Construction of New Lattices
- Sub- and Superlattices and Quotients
sub<L | S>: Lat, List → Latext< L | S >: Lat, List → LatT * L: AlgMatElt, Lat → LatT * L: ModMatRng, Lat → Lats * L: RngElt, Lat → LatL * s: Lat, RngElt → LatJ * L: RngInt, Lat → LatJ * L: RngIntFracIdl, Lat → LatL / s: Lat, RngElt → Latquo< L | S >: Lat, List → GrpAb, MapL / S: Lat, Lat → GrpAb, MapElementaryDivisors(A, B): Lat, Lat → [ RngIntFracIdl ]Discriminant(A, B): Lat, Lat → RngIntFracIdlIndex(L, S): Lat, Lat → RngIntExample: Sub Super Quo
- Standard Constructions of New Lattices
Dual(L): Lat → LatPartialDual(L, n): Lat, RngIntElt → LatDualBasisLattice(L): Lat → LatDualQuotient(L): Lat → GrpAb, Lat, MapEvenSublattice(L): Lat → Lat, MapExample: dualL + M: Lat, Lat → LatL meet M: Lat, Lat → LatDirectSum(L, M): Lat, Lat → LatOrthogonalSum(L, M): Lat, Lat → LatOrthogonalDecomposition(L): Lat → [Lat]OrthogonalDecomposition(L, F): Lat, [Mtrx] → [Lat]OrthogonalDecomposition(F): [Mtrx] → [* Mtrx *], [* [Mtrx] *]TensorProduct(L, M): Lat, Lat → LatExteriorSquare(L): Lat → LatSymmetricSquare(L): Lat → LatPureLattice(L): Lat → LatIntegralBasisLattice(L): Lat → Lat, RngIntElt
- Reduction of Matrices and Lattices
- LLL Reduction
Example: LLL UsageLLL(X): ModMatRngElt → ModMatRngElt, AlgMatElt, RngIntEltLLL(X): AlgMatElt → AlgMatElt, AlgMatElt, RngIntEltBasisReduction(X): ModMatRngElt → ModMatRngElt, AlgMatElt, RngIntEltBasisReduction(X): AlgMatElt → AlgMatElt, AlgMatElt, RngIntEltLLLGram(F): ModMatRngElt → ModMatRngElt, AlgMatElt, RngIntEltLLLGram(F): AlgMatElt → AlgMatElt, AlgMatElt, RngIntEltLLLBasisMatrix(L): Lat → ModMatElt, AlgMatEltLLLGramMatrix(L): Lat → AlgMatElt, AlgMatEltLLL(L): Lat → Lat, AlgMatEltBasisReduction(L): Lat → Lat, AlgMatEltSetVerbose("LLL", v): MonStgElt, RngIntEltExample: LLLXGCD
- Pair Reduction
- Seysen Reduction
- HKZ Reduction
- BKZ Reduction
- Minkowski Reduction
- Greedy Reduction
- Recovering a Short Basis from Short Lattice Vectors
- Minima and Element Enumeration
- Minimum, Density and Kissing Number
- Shortest and Closest Vectors
ShortestVectors(L): Lat → [ LatElt ]ShortestVectorsMatrix(L): Lat → ModMatRngEltShortestVector(L): Lat → LatEltClosestVectors(L, w): Lat, ModTupRngElt → [ LatElt ], RngEltClosestVectors(L, w): Lat, LatElt → [ LatElt ], RngEltClosestVectorsMatrix(L, w): Lat, ModTupRngElt → ModMatRngElt, RngEltClosestVectorsMatrix(L, w): Lat, LatElt → ModMatRngElt, RngEltClosestVector(L, w): Lat, ModTupRngElt → LatEltClosestVector(L, w): Lat, LatElt → LatEltExample: Closest
- Short and Close Vectors
ShortVectors(L, u): Lat, RngElt → [ <LatElt, RngElt> ]ShortVectors(L, l, u): Lat, RngElt, RngElt → [ <LatElt, RngElt> ]ShortVectorsMatrix(L, u): Lat, RngElt → ModMatRngEltShortVectorsMatrix(L, l, u): Lat, RngElt, RngElt → ModMatRngEltShortVector(L, u): Lat, RngElt → LatEltShortVector(L, l, u): Lat, RngElt, RngElt → LatEltCloseVectors(L, w, u): Lat, ModTupRngElt, RngElt → [ <LatElt, RngElt> ]CloseVectors(L, w, l, u): Lat, ModTupRngElt, RngElt, RngElt → [ <LatElt, RngElt> ]CloseVectors(L, w, u): Lat, LatElt, RngElt → [ <LatElt, RngElt> ]CloseVectors(L, w, l, u): Lat, LatElt, RngElt, RngElt → [ <LatElt, RngElt> ]CloseVectorsMatrix(L, w, u): Lat, ModTupRngElt, RngElt → ModMatRngEltCloseVectorsMatrix(L, w, l, u): Lat, ModTupRngElt, RngElt, RngElt → ModMatRngEltCloseVectorsMatrix(L, w, u): Lat, LatElt, RngElt → ModMatRngEltCloseVectorsMatrix(L, l, u): Lat, LatElt, RngElt, RngElt → ModMatRngEltCloseVector(L, w, u): Lat, ModTupRngElt, RngElt → LatEltCloseVector(L, w, l, u): Lat, ModTupRngElt, RngElt, RngElt → LatEltCloseVector(L, w, u): Lat, LatElt, RngElt → LatEltCloseVector(L, w, l, u): Lat, LatElt, RngElt, RngElt → LatEltExample: KnapsackExample: Singular Elements
- Short and Close Vector Processes
ShortVectorsProcess(L, u): Lat, RngElt → LatEnumProcShortVectorsProcess(L, l, u): Lat, RngElt, RngElt → LatEnumProcCloseVectorsProcess(L, w, u): Lat, ModTupRngElt, RngElt → LatEnumProcCloseVectorsProcess(L, w, l, u): Lat, ModTupRngElt, RngElt, RngElt → LatEnumProcCloseVectorsProcess(L, w, u): Lat, LatElt, RngElt → LatEnumProcCloseVectorsProcess(L, w, l, u): Lat, LatElt, RngElt, RngElt → LatEnumProcNextVector(P): LatEnumProc → LatElt, RngEltIsEmpty(P): LatEnumProc → BoolElt
- Successive Minima and Theta Series
- Lattice Enumeration Utilities
- Theta Series as Modular Forms
- Voronoi Cells, Holes and Covering Radius
- Orthogonalization
Orthogonalize(M): MtrxSpcElt → MtrxSpcElt, AlgMatElt, RngIntEltOrthogonalization(F): MtrxSpcElt → MtrxSpcElt, AlgMatElt, RngIntEltOrthogonalizeGram(F): MtrxSpcElt → MtrxSpcElt, AlgMatElt, RngIntEltOrthogonalizeGram(F, a): MtrxSpcElt, FldAut → MtrxSpcElt, AlgMatElt, RngIntEltOrthogonalize(L): Lat → Lat, AlgMatEltOrthonormalize(M, K): MtrxSpcElt, Fld → AlgMatEltCholesky(M, K): MtrxSpcElt, Fld → AlgMatEltOrthonormalize(M): MtrxSpcElt → AlgMatEltCholesky(M): MtrxSpcElt → AlgMatEltOrthonormalize(L, K): Lat, FldRe → AlgMatEltCholesky(L, K): Lat, FldRe → AlgMatEltOrthonormalize(L): Lat → AlgMatEltCholesky(L): Lat → AlgMatEltExample: Orthogonalize
- Testing Matrices for Definiteness
- Genera and Spinor Genera
- Genus Constructions
Genus(L): Lat → SymGenGenus(G): SymGen → SymGenGenera(r, s, d): RngIntElt, RngIntElt, RngIntElt → [ SymGen ]SpinorGenus(L): Lat → SymGenSpinorGenera(G): SymGen → [ SymGen ]Genus(r, s, d, S): RngIntElt, RngIntElt, RngIntElt, [ SymGenLoc ] → SymGen
- Local Genus Constructions
LocalGenus(L, p): Lat, RngIntElt → SymGenLocLocalGenus(A, p): AlgMatElt[RngInt], RngIntElt → SymGenLocCanonicalLocalGenus(G): SymGenLoc → SymGenLocLocalGenus(p, v, r, d): RngIntElt, [ RngIntElt ], [ RngIntElt ], [ RngElt ] → SymGenLocLocalGenera(n, d, p): RngIntElt, RngIntElt, RngIntElt → [ SymGenLoc ]pAdicDiagonalization(L, p): Lat, RngIntElt → LatJordanDecomposition(L, p): Lat, RngIntElt → List, List, SeqEnumJordanDecomposition(L, P): Lat, RngInt → List, List, SeqEnumpAdicJordanDecomposition(L, p): Lat, RngIntElt → AlgMatElt, AlgMatEltLocalModification(L, G, p): Lat, AlgMatElt[RngInt], RngIntElt → Lat
- Invariants of Genera and Spinor Genera
- Invariants of \(p\)-adic Genera
- Neighbour Relations and Graphs
Neighbour(L, v, p): Lat, LatElt, RngIntElt → LatNeighbor(L, v, p): Lat, LatElt, RngIntElt → LatNeighbours(L, p): Lat, RngIntElt → LatNeighbors(L, p): Lat, RngIntElt → LatNeighbourClosure(L, p): Lat, RngIntElt → LatNeighborClosure(L, p): Lat, RngIntElt → LatGenusRepresentatives(L): Lat → [ Lat ], AssocSpinorRepresentatives(L): Lat → [ Lat ], AssocRepresentatives(G): SymGen → [ Lat ], AssocRepresentatives(G): SymGen → [ LatNF ], AssocPositiveDefiniteQuadraticLattices(n, d): RngIntElt, RngIntElt → [ [ Lat ] ]TernaryQuadraticLattices(d): RngIntElt → [ [ Lat ] ]QuaternaryQuadraticLattices(d): RngIntElt → [ [ Lat ] ]QuinaryQuadraticLattices(d): RngIntElt → [ [ Lat ] ]AdjacencyMatrix(G, p): SymGen, RngIntElt → AlgMatEltExample: NeighbourExample: Genus
- Attributes of Lattices
- Database of Lattices
- Creating the Database
- Database Information
# D: DB → RngIntEltNumberOfLattices(D): DB → RngIntEltLargestDimension(D): DB → RngIntEltNumberOfLattices(D, d): DB, RngIntElt → RngIntEltNumberOfLattices(D, N): DB, MonStgElt → RngIntEltLatticeName(D, i): DB, RngIntElt → MonStgElt, RngIntEltLatticeName(D, d, i): DB, RngIntElt, RngIntElt → RecMonStgElt, RngIntEltLatticeName(D, N): DB, MonStgElt → RecMonStgElt, RngIntEltLatticeName(D, N, i): DB, MonStgElt, RngIntElt → RecMonStgElt, RngIntEltExample: Latdb Names
- Accessing the Database
Lattice(D, i: parameters): DB, RngIntElt → LatticeLattice(D, d, i: parameters): DB, RngIntElt, RngIntElt → LatticeLattice(D, N: parameters): DB, MonStgElt → LatticeLattice(D, N, i: parameters): DB, MonStgElt, RngIntElt → LatticeLatticeData(D, i): DB, RngIntElt → RecLatticeData(D, d, i): DB, RngIntElt, RngIntElt → RecLatticeData(D, N): DB, MonStgElt → RecLatticeData(D, N, i): DB, MonStgElt, RngIntElt → RecExample: latdb
- Hermitian Lattices
