  
  
                                     [1m[4m[31m[1mGUAVA[1m[4m[31m[0m
  
  
           [1m[4m[31mA [1mGAP[1m[4m[31m4 Package for computing with error-correcting codes  [0m
  
  
                                  Version 2.7
  
  
                                  May 23, 2006
  
  
                               Jasper Cramwinckel
  
                                Erik Roijackers
  
                                 Reinald Baart
  
                                  Eric Minkes
  
                                   Lea Ruscio
  
                                  Cen Tjhai  
  
                          David Joyner (Maintainer)  
  
  
  
  Cen Tjhai  
      Email:    [34mmailto:cen.tjhai@plymouth.ac.uk[0m
      Homepage: [34mhttp://www.plymouth.ac.uk/staff/ctjhai[0m
      Address:  School of Computing, Communications and Electronics,
                University of Plymouth,
                Plymouth, Devon, PL4 8AA, UK.
  
  
  David Joyner (Maintainer)  
      Email:    [34mmailto: wdj@usna.edu[0m
      Homepage: [34mhttp://cadigweb.ew.usna.edu/~wdj/gap/GUAVA/[0m
      Address:  Mathematics Department,
                U. S. Naval Academy,
                Annapolis, MD,
                21402 USA.
  
  
  
  -------------------------------------------------------
  [1m[4m[31mCopyright[0m
  (C)  1992-2003  Jasper  Cramwinckel,  Erik  Roijackers,Reinald  Baart,  Eric
  Minkes,  Lea  Ruscio (for the tex version) (C) 2004 David Joyner, Cen Tjhai,
  Jasper Cramwinckel, Erik Roijackers, Reinald Baart, Eric Minkes, Lea Ruscio.
  
  [1mGUAVA[0m  is  released under the GNU General Public License (GPL). This file is
  part  of  [1mGUAVA[0m,  though  as documentation it is released under the GNU Free
  Documentation License (see [34mhttp://www.gnu.org/licenses/licenses.html#FDL[0m).
  
  [1mGUAVA[0m  is  free software; you can redistribute it and/or modify it under the
  terms  of  the  GNU General Public License as published by the Free Software
  Foundation;  either  version 2 of the License, or (at your option) any later
  version.
  
  [1mGUAVA[0m  is  distributed  in  the hope that it will be useful, but WITHOUT ANY
  WARRANTY;  without  even  the implied warranty of MERCHANTABILITY or FITNESS
  FOR  A  PARTICULAR  PURPOSE.  See  the  GNU  General Public License for more
  details.
  
  You should have received a copy of the GNU General Public License along with
  [1mGUAVA[0m; if not, write to the Free Software Foundation, Inc., 59 Temple Place,
  Suite 330, Boston, MA 02111-1307 USA
  
  For more details, see [34mhttp://www.fsf.org/licenses/gpl.html[0m.
  
  For  many  years  [1mGUAVA[0m  has been released along with the ``backtracking'' C
  programs of J. Leon. In one of his *.c files the following statements occur:
  ``Copyright  (C)  1992  by Jeffrey S. Leon. This software may be used freely
  for  educational  and  research  purposes. Any other use requires permission
  from the author.''
  
  
  -------------------------------------------------------
  [1m[4m[31mAcknowledgements[0m
  [1mGUAVA[0m  was  originally  written  by Jasper Cramwinckel, Erik Roijackers, and
  Reinald  Baart  in  the  early-to-mid 1990's as a final project during their
  study  of  Mathematics  at the Delft University of Technology, Department of
  Pure  Mathematics,  under  the  direction of Professor Juriaan Simonis. This
  work was continued in Aachen, at Lehrstuhl D fur Mathematik. In version 1.3,
  new  functions  were  added  by  Eric  Minkes, also from Delft University of
  Technology.
  
  JC,  ER  and  RB  would  like to thank the [1mGAP[0m people at the RWTH Aachen for
  their  support,  A.E.  Brouwer  for  his  advice  and  J.  Simonis  for  his
  supervision.
  
  The  [1mGAP[0m 4 version of [1mGUAVA[0m (versions 1.4 and 1.5) was created by Lea Ruscio
  and (since 2001, starting with version 1.6) is currently maintained by David
  Joyner,  who  (with  the  help  of  several  students) has added several new
  functions.  Starting  with version 2.7, the ``best linear code'' tables have
  been  updated.  For  further  details,  see  the  CHANGES  file in the [1mGUAVA[0m
  directory,                 also                 available                 at
  [34mhttp://cadigweb.ew.usna.edu/~wdj/gap/GUAVA/CHANGES.guava[0m.
  
  This  documentation  was  prepared with the [1mGAPDoc[0m package of Frank L\"ubeck
  and  Max  Neunh\"offer.  The conversion from TeX to [1mGAPDoc[0m's XML was done by
  David Joyner in 2004.
  
  Please  send  bug  reports,  suggestions  and  other comments about [1mGUAVA[0m to
  [34mmailto:support@gap-system.org[0m.  Currently  known  bugs  and  suggested [1mGUAVA[0m
  projects    are    listed    on    the    bugs   and   projects   web   page
  [34mhttp://cadigweb.ew.usna.edu/~wdj/gap/GUAVA/guava2do.html[0m. Older releases and
  further    history    can    be    found    on    the    [1mGUAVA[0m    web   page
  [34mhttp://cadigweb.ew.usna.edu/~wdj/gap/GUAVA/[0m.
  
  [22m[36mContributors[0m: Other than the authors listed on the title page, the following
  people  have  contributed code to the [1mGUAVA[0m project: Alexander Hulpke, Steve
  Linton,  Frank  L\"ubeck,  Aron  Foster,  Wayne  Irons, Clifton (``Clipper")
  Lennon,  Jason  McGowan,  Shuhong Gao, Greg Gamble, and, indirectly, Jeffrey
  Leon.
  
  For  documentation  on Leon's programs, see the src/leon/doc subdirectory of
  [1mGUAVA[0m.
  
  
  -------------------------------------------------------
  
  
  [1m[4m[31mContent (guava)[0m
  
  1. Introduction
    1.1 Introduction to the [1mGUAVA[0m package
    1.2 Installing [1mGUAVA[0m
    1.3 Loading [1mGUAVA[0m
  2. Coding theory functions in [1mGAP[0m
    2.1 Distance functions
      2.1-1 AClosestVectorCombinationsMatFFEVecFFE
      2.1-2 AClosestVectorComb..MatFFEVecFFECoords
      2.1-3 DistancesDistributionMatFFEVecFFE
      2.1-4 DistancesDistributionVecFFEsVecFFE
      2.1-5 WeightVecFFE
      2.1-6 DistanceVecFFE
    2.2 Other functions
      2.2-1 ConwayPolynomial
      2.2-2 RandomPrimitivePolynomial
  3. Codewords
    3.1 Construction of Codewords
      3.1-1 Codeword
      3.1-2 CodewordNr
      3.1-3 IsCodeword
    3.2 Comparisons of Codewords
      3.2-1 =
    3.3 Arithmetic Operations for Codewords
      3.3-1 +
      3.3-2 -
      3.3-3 +
    3.4 Functions that Convert Codewords to Vectors or Polynomials
      3.4-1 VectorCodeword
      3.4-2 PolyCodeword
    3.5 Functions that Change the Display Form of a Codeword
      3.5-1 TreatAsVector
      3.5-2 TreatAsPoly
    3.6 Other Codeword Functions
      3.6-1 NullWord
      3.6-2 DistanceCodeword
      3.6-3 Support
      3.6-4 WeightCodeword
  4. Codes
    4.1 Comparisons of Codes
      4.1-1 =
    4.2 Operations for Codes
      4.2-1 +
      4.2-2 *
      4.2-3 *
      4.2-4 InformationWord
    4.3 Boolean Functions for Codes
      4.3-1 in
      4.3-2 IsSubset
      4.3-3 IsCode
      4.3-4 IsLinearCode
      4.3-5 IsCyclicCode
      4.3-6 IsPerfectCode
      4.3-7 IsMDSCode
      4.3-8 IsSelfDualCode
      4.3-9 IsSelfOrthogonalCode
      4.3-10 IsSelfComplementaryCode
      4.3-11 IsAffineCode
      4.3-12 IsAlmostAffineCode
    4.4 Equivalence and Isomorphism of Codes
      4.4-1 IsEquivalent
      4.4-2 CodeIsomorphism
      4.4-3 AutomorphismGroup
      4.4-4 PermutationAutomorphismGroup
    4.5 Domain Functions for Codes
      4.5-1 IsFinite
      4.5-2 Size
      4.5-3 LeftActingDomain
      4.5-4 Dimension
      4.5-5 AsSSortedList
    4.6 Printing and Displaying Codes
      4.6-1 Print
      4.6-2 String
      4.6-3 Display
      4.6-4 DisplayBoundsInfo
    4.7 Generating (Check) Matrices and Polynomials
      4.7-1 GeneratorMat
      4.7-2 CheckMat
      4.7-3 GeneratorPol
      4.7-4 CheckPol
      4.7-5 RootsOfCode
    4.8 Parameters of Codes
      4.8-1 WordLength
      4.8-2 Redundancy
      4.8-3 MinimumDistance
      4.8-4 MinimumDistanceLeon
      4.8-5 DecreaseMinimumDistanceUpperBound
      4.8-6 MinimumDistanceRandom
      4.8-7 CoveringRadius
      4.8-8 SetCoveringRadius
    4.9 Distributions
      4.9-1 MinimumWeightWords
      4.9-2 WeightDistribution
      4.9-3 InnerDistribution
      4.9-4 DistancesDistribution
      4.9-5 OuterDistribution
    4.10 Decoding Functions
      4.10-1 Decode
      4.10-2 Decodeword
      4.10-3 GeneralizedReedSolomonDecoderGao
      4.10-4 GeneralizedReedSolomonListDecoder
      4.10-5 BitFlipDecoder
      4.10-6 NearestNeighborGRSDecodewords
      4.10-7 NearestNeighborDecodewords
      4.10-8 Syndrome
      4.10-9 SyndromeTable
      4.10-10 StandardArray
      4.10-11 PermutationDecode
      4.10-12 PermutationDecodeNC
  5. Generating Codes
    5.1 Generating Unrestricted Codes
      5.1-1 ElementsCode
      5.1-2 HadamardCode
      5.1-3 ConferenceCode
      5.1-4 MOLSCode
      5.1-5 RandomCode
      5.1-6 NordstromRobinsonCode
      5.1-7 GreedyCode
      5.1-8 LexiCode
    5.2 Generating Linear Codes
      5.2-1 GeneratorMatCode
      5.2-2 CheckMatCodeMutable
      5.2-3 CheckMatCode
      5.2-4 HammingCode
      5.2-5 ReedMullerCode
      5.2-6 AlternantCode
      5.2-7 GoppaCode
      5.2-8 GeneralizedSrivastavaCode
      5.2-9 SrivastavaCode
      5.2-10 CordaroWagnerCode
      5.2-11 FerreroDesignCode
      5.2-12 RandomLinearCode
      5.2-13 OptimalityCode
      5.2-14 BestKnownLinearCode
    5.3 Gabidulin Codes
      5.3-1 GabidulinCode
      5.3-2 EnlargedGabidulinCode
      5.3-3 DavydovCode
      5.3-4 TombakCode
      5.3-5 EnlargedTombakCode
    5.4 Golay Codes
      5.4-1 BinaryGolayCode
      5.4-2 ExtendedBinaryGolayCode
      5.4-3 TernaryGolayCode
      5.4-4 ExtendedTernaryGolayCode
    5.5 Generating Cyclic Codes
      5.5-1 GeneratorPolCode
      5.5-2 CheckPolCode
      5.5-3 RootsCode
      5.5-4 BCHCode
      5.5-5 ReedSolomonCode
      5.5-6 QRCode
      5.5-7 QQRCodeNC
      5.5-8 QQRCode
      5.5-9 FireCode
      5.5-10 WholeSpaceCode
      5.5-11 NullCode
      5.5-12 RepetitionCode
      5.5-13 CyclicCodes
      5.5-14 NrCyclicCodes
    5.6 Evaluation Codes
      5.6-1 EvaluationCode
      5.6-2 GeneralizedReedSolomonCode
      5.6-3 GeneralizedReedMullerCode
      5.6-4 ToricPoints
      5.6-5 ToricCode
    5.7 Algebraic geometric codes
      5.7-1 AffineCurve
      5.7-2 AffinePointsOnCurve
      5.7-3 GenusCurve
      5.7-4 GOrbitPoint 
      5.7-5 DivisorOnAffineCurve
      5.7-6 DivisorAddition 
      5.7-7 DivisorDegree 
      5.7-8 DivisorNegate 
      5.7-9 DivisorIsZero 
      5.7-10 DivisorsEqual 
      5.7-11 DivisorGCD 
      5.7-12 DivisorLCM 
      5.7-13 RiemannRochSpaceBasisFunctionP1 
      5.7-14 DivisorOfRationalFunctionP1 
      5.7-15 RiemannRochSpaceBasisP1 
      5.7-16 MoebiusTransformation 
      5.7-17 ActionMoebiusTransformationOnFunction 
      5.7-18 ActionMoebiusTransformationOnDivisorP1 
      5.7-19 IsActionMoebiusTransformationOnDivisorDefinedP1 
      5.7-20 DivisorAutomorphismGroupP1 
      5.7-21 MatrixRepresentationOnRiemannRochSpaceP1 
      5.7-22 GoppaCodeClassical
      5.7-23 EvaluationBivariateCode
      5.7-24 EvaluationBivariateCodeNC
      5.7-25 OnePointAGCode
  6. Manipulating Codes
    6.1 Functions that Generate a New Code from a Given Code
      6.1-1 ExtendedCode
      6.1-2 PuncturedCode
      6.1-3 EvenWeightSubcode
      6.1-4 PermutedCode
      6.1-5 ExpurgatedCode
      6.1-6 AugmentedCode
      6.1-7 RemovedElementsCode
      6.1-8 AddedElementsCode
      6.1-9 ShortenedCode
      6.1-10 LengthenedCode
      6.1-11 ResidueCode
      6.1-12 ConstructionBCode
      6.1-13 DualCode
      6.1-14 ConversionFieldCode
      6.1-15 TraceCode
      6.1-16 CosetCode
      6.1-17 ConstantWeightSubcode
      6.1-18 StandardFormCode
      6.1-19 PiecewiseConstantCode
    6.2 Functions that Generate a New Code from Two Given Codes
      6.2-1 DirectSumCode
      6.2-2 UUVCode
      6.2-3 DirectProductCode
      6.2-4 IntersectionCode
      6.2-5 UnionCode
      6.2-6 ExtendedDirectSumCode
      6.2-7 AmalgamatedDirectSumCode
      6.2-8 BlockwiseDirectSumCode
  7. Bounds on codes, special matrices and miscellaneous functions
    7.1 Distance bounds on codes
      7.1-1 UpperBoundSingleton
      7.1-2 UpperBoundHamming
      7.1-3 UpperBoundJohnson
      7.1-4 UpperBoundPlotkin
      7.1-5 UpperBoundElias
      7.1-6 UpperBoundGriesmer
      7.1-7 IsGriesmerCode
      7.1-8 UpperBound
      7.1-9 LowerBoundMinimumDistance
      7.1-10 LowerBoundGilbertVarshamov
      7.1-11 LowerBoundSpherePacking
      7.1-12 UpperBoundMinimumDistance
      7.1-13 BoundsMinimumDistance
    7.2 Covering radius bounds on codes
      7.2-1 BoundsCoveringRadius
      7.2-2 IncreaseCoveringRadiusLowerBound
      7.2-3 ExhaustiveSearchCoveringRadius
      7.2-4 GeneralLowerBoundCoveringRadius
      7.2-5 GeneralUpperBoundCoveringRadius
      7.2-6 LowerBoundCoveringRadiusSphereCovering
      7.2-7 LowerBoundCoveringRadiusVanWee1
      7.2-8 LowerBoundCoveringRadiusVanWee2
      7.2-9 LowerBoundCoveringRadiusCountingExcess
      7.2-10 LowerBoundCoveringRadiusEmbedded1
      7.2-11 LowerBoundCoveringRadiusEmbedded2
      7.2-12 LowerBoundCoveringRadiusInduction
      7.2-13 UpperBoundCoveringRadiusRedundancy
      7.2-14 UpperBoundCoveringRadiusDelsarte
      7.2-15 UpperBoundCoveringRadiusStrength
      7.2-16 UpperBoundCoveringRadiusGriesmerLike
      7.2-17 UpperBoundCoveringRadiusCyclicCode
    7.3 Special matrices in [1mGUAVA[0m
      7.3-1 KrawtchoukMat
      7.3-2 GrayMat
      7.3-3 SylvesterMat
      7.3-4 HadamardMat
      7.3-5 VandermondeMat
      7.3-6 PutStandardForm
      7.3-7 IsInStandardForm
      7.3-8 PermutedCols
      7.3-9 VerticalConversionFieldMat
      7.3-10 HorizontalConversionFieldMat
      7.3-11 MOLS
      7.3-12 IsLatinSquare
      7.3-13 AreMOLS
    7.4 Some functions related to the norm of a code
      7.4-1 CoordinateNorm
      7.4-2 CodeNorm
      7.4-3 IsCoordinateAcceptable
      7.4-4 GeneralizedCodeNorm
      7.4-5 IsNormalCode
    7.5 Miscellaneous functions
      7.5-1 CodeWeightEnumerator
      7.5-2 CodeDistanceEnumerator
      7.5-3 CodeMacWilliamsTransform
      7.5-4 CodeDensity
      7.5-5 SphereContent
      7.5-6 Krawtchouk
      7.5-7 PrimitiveUnityRoot
      7.5-8 PrimitivePolynomialsNr
      7.5-9 IrreduciblePolynomialsNr
      7.5-10 MatrixRepresentationOfElement
      7.5-11 ReciprocalPolynomial
      7.5-12 CyclotomicCosets
      7.5-13 WeightHistogram
      7.5-14 MultiplicityInList
      7.5-15 MostCommonInList
      7.5-16 RotateList
      7.5-17 CirculantMatrix
    7.6 Miscellaneous polynomial functions
      7.6-1 MatrixTransformationOnMultivariatePolynomial 
      7.6-2 DegreeMultivariatePolynomial
      7.6-3 DegreesMultivariatePolynomial
      7.6-4 CoefficientMultivariatePolynomial
      7.6-5 SolveLinearSystem
      7.6-6 GuavaVersion
      7.6-7 CoefficientToPolynomial
      7.6-8 CoefficientToPolynomial
      7.6-9 DegreesMonomialTerm
      7.6-10 DivisorsMultivariatePolynomial
  
  
  -------------------------------------------------------
