  
  
                                     [1X ALCO [101X
  
  
                      [1X Tools for algebraic combinatorics [101X
  
  
                                     1.1.2
  
  
                                5 September 2025
  
  
                                Benjamin Nasmith
  
  
  
  Benjamin Nasmith
      Email:    [7Xmailto:bnasmith@proton.me[107X
      Homepage: [7Xhttps://github.com/BNasmith/[107X
  
  -------------------------------------------------------
  [1XAbstract[101X
  [33X[0;0YALCO  provides implementations in [5XGAP[105X of octonion algebras, Jordan algebras,
  and  certain  important integer subrings of those algebras. It also provides
  tools  to  compute  the  parameters of t-designs in spherical and projective
  spaces  (modeled  as  manifolds of primitive idempotent elements in a simple
  Euclidean  Jordan  algebra). Finally, this package provides tools to explore
  octonion lattice constructions, including octonion Leech lattices.[133X
  
  
  -------------------------------------------------------
  [1XCopyright[101X
  [33X[0;0Y© 2024 by Benjamin Nasmith[133X
  
  [33X[0;0YThis  package  may  be distributed under the terms and conditions of the GNU
  Public License Version 3 or (at your option) any later version.[133X
  
  
  -------------------------------------------------------
  [1XAcknowledgements[101X
  [33X[0;0YThis documentation was prepared using the [5XGAPDoc[105X package.[133X
  
  
  -------------------------------------------------------
  
  
  [1XContents (ALCO)[101X
  
  1 [33X[0;0YIntroduction[133X
  2 [33X[0;0YOctonions[133X
    2.1 [33X[0;0YOctonion Algebras[133X
      2.1-1 [33X[0;0YOctonion Filters[133X
      2.1-2 OctonionAlgebra
      2.1-3 [33X[0;0YOctavian Integers[133X
      2.1-4 OctonionE8Basis
    2.2 [33X[0;0YProperties of Octonions[133X
      2.2-1 Norm
      2.2-2 Trace
      2.2-3 ComplexConjugate
      2.2-4 RealPart
    2.3 [33X[0;0YOther Octonion Tools[133X
      2.3-1 [33X[0;0YConverting Octonion Vectors[133X
      2.3-2 VectorToIdempotentMatrix
      2.3-3 WeylReflection
    2.4 [33X[0;0YQuaternion Tools[133X
      2.4-1 Norm
      2.4-2 Trace
      2.4-3 [33X[0;0YHurwitz Integers[133X
      2.4-4 QuaternionD4Basis
    2.5 [33X[0;0YIcosian Tools[133X
      2.5-1 [33X[0;0YIcosian Ring[133X
      2.5-2 IcosianH4Generators
      2.5-3 [33X[0;0YGoldenModSigma[133X
    2.6 [33X[0;0YOther Integer Rings[133X
      2.6-1 EisensteinIntegers
      2.6-2 KleinianIntegers
  3 [33X[0;0YSimple Euclidean Jordan Algebras[133X
    3.1 [33X[0;0YFilters and Basic Attributes[133X
      3.1-1 [33X[0;0YJordan Filters[133X
      3.1-2 [33X[0;0YJordan Rank[133X
      3.1-3 [33X[0;0YJordan Degree[133X
      3.1-4 Trace
      3.1-5 Determinant
      3.1-6 Norm
      3.1-7 GenericMinimalPolynomial
    3.2 [33X[0;0YJordan Algebra Constructions[133X
      3.2-1 SimpleEuclideanJordanAlgebra
      3.2-2 JordanSpinFactor
      3.2-3 HermitianSimpleJordanAlgebra
      3.2-4 JordanHomotope
    3.3 [33X[0;0YThe Albert Algebra[133X
      3.3-1 AlbertAlgebra
      3.3-2 AlbertVectorToHermitianMatrix
      3.3-3 HermitianMatrixToAlbertVector
    3.4 [33X[0;0YThe Quadratic Representation[133X
      3.4-1 JordanQuadraticOperator
      3.4-2 JordanTripleSystem
    3.5 [33X[0;0YAdditional Tools and Properties[133X
      3.5-1 HermitianJordanAlgebraBasis
      3.5-2 JordanMatrixBasis
      3.5-3 HermitianMatrixToJordanVector
      3.5-4 JordanAlgebraGramMatrix
      3.5-5 JordanAdjugate
      3.5-6 IsPositiveDefinite
  4 [33X[0;0YSpherical and Projective Designs[133X
    4.1 [33X[0;0YJacobi Polynomials[133X
      4.1-1 JacobiPolynomial
      4.1-2 [33X[0;0YRenormalized Jacobi Polynomials[133X
    4.2 [33X[0;0YJordan Designs[133X
      4.2-1 [33X[0;0YJordan Design Categories[133X
      4.2-2 JordanDesignByParameters
      4.2-3 [33X[0;0YJordan Rank and Degree[133X
      4.2-4 JordanDesignQPolynomials
      4.2-5 JordanDesignConnectionCoefficients
    4.3 [33X[0;0YDesigns with an Angle Set[133X
      4.3-1 IsJordanDesignWithAngleSet
      4.3-2 [33X[0;0YDesign Angle Sets[133X
      4.3-3 JordanDesignByAngleSet
      4.3-4 JordanDesignNormalizedAnnihilatorPolynomial
      4.3-5 JordanDesignNormalizedIndicatorCoefficients
      4.3-6 IsJordanDesignWithPositiveIndicatorCoefficients
      4.3-7 JordanDesignSpecialBound
    4.4 [33X[0;0YDesigns with Angle Set and Cardinality[133X
      4.4-1 [33X[0;0YDesign Cardinality[133X
      4.4-2 [33X[0;0YDesigns at the Special Bound[133X
      4.4-3 JordanDesignAnnihilatorPolynomial
      4.4-4 JordanDesignIndicatorCoefficients
      4.4-5 [33X[0;0YDesign Strength[133X
      4.4-6 [33X[0;0YSchemes and Tight Designs[133X
    4.5 [33X[0;0YDesigns Admitting a Regular Scheme[133X
      4.5-1 JordanDesignSubdegrees
    4.6 [33X[0;0YDesigns Admitting an Association Scheme[133X
      4.6-1 JordanDesignBoseMesnerAlgebra
      4.6-2 JordanDesignBoseMesnerIdempotentBasis
      4.6-3 JordanDesignIntersectionNumbers
      4.6-4 JordanDesignKreinNumbers
      4.6-5 JordanDesignFirstEigenmatrix
      4.6-6 JordanDesignSecondEigenmatrix
      4.6-7 JordanDesignMultiplicities
      4.6-8 DesignValencies
      4.6-9 JordanDesignReducedAdjacencyMatrices
    4.7 [33X[0;0YExamples[133X
  5 [33X[0;0YOctonion Lattice Constructions[133X
    5.1 [33X[0;0YGram Matrices and Octonion Lattices[133X
      5.1-1 IsLeechLatticeGramMatrix
      5.1-2 IsGossetLatticeGramMatrix
      5.1-3 [33X[0;0YMiracle Octad Generator (MOG) Coordinates[133X
      5.1-4 IsOctonionLattice
      5.1-5 OctonionLatticeByGenerators
    5.2 [33X[0;0YOctonion Lattice Attributes[133X
      5.2-1 UnderlyingOctonionRing
      5.2-2 OctonionGramMatrix
      5.2-3 GeneratorsAsCoefficients
      5.2-4 LLLReducedBasisCoefficients
      5.2-5 [33X[0;0YOctonion Lattice Dimension[133X
      5.2-6 GramMatrix
      5.2-7 [33X[0;0YLattice Basis[133X
    5.3 [33X[0;0YOctonion Lattice Operations[133X
      5.3-1 \in
      5.3-2 ScalarProduct
      5.3-3 [33X[0;0YSublattice Identification[133X
      5.3-4 [33X[0;0YLattice Vector Coefficients[133X
  6 [33X[0;0YClosure Tools[133X
    6.1 [33X[0;0YBrute Force Method[133X
      6.1-1 Closure
    6.2 [33X[0;0YRandom Choice Methods[133X
      6.2-1 RandomElementClosure
      6.2-2 RandomOrbitOnSets
  
  
  [32X
