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# GUAVA

## A GAP4 Package for computing with error-correcting codes

Version 2.4

July 4, 2005

Jasper Cramwinckel

Erik Roijackers

Reinald Baart

Eric Minkes

Lea Ruscio

David Joyner
e-mail: wdj@usna.edu
WWW: http://cadigweb.ew.usna.edu/~wdj/gap/GUAVA/
Address:
Mathematics Department,
U. S. Naval Academy,
Annapolis, MD,
21402 USA.

### Copyright

(C) 1992-2003 Jasper Cramwinckel, Erik Roijackers,Reinald Baart, Eric Minkes, Lea Ruscio (for the tex version) (C) 2004 David Joyner, Jasper Cramwinckel, Erik Roijackers, Reinald Baart, Eric Minkes, Lea Ruscio.

GUAVA is released under the GNU General Public License (GPL). This file is part of GUAVA, though as documentation it is released under the GNU Free Documentation License (see http://www.gnu.org/licenses/licenses.html#FDL).

GUAVA 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.

GUAVA 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 GUAVA; if not, write to the Free Software Foundation, Inc., 59 Temple Place, Suite 330, Boston, MA 02111-1307 USA

For more details, see http://www.fsf.org/licenses/gpl.html.

For many years GUAVA 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.''

### Acknowledgements

GUAVA 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 GAP people at the RWTH Aachen for their support, A.E. Brouwer for his advice and J. Simonis for his supervision.

The GAP 4 version of GUAVA (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. For further details, see the CHANGES file in the GUAVA directory, also available at http://cadigweb.ew.usna.edu/~wdj/gap/GUAVA/CHANGES.guava.

This documentation was prepared with the GAPDoc package of Frank L\"ubeck and Max Neunh\"offer. The conversion from TeX to GAPDoc's XML was done by David Joyner in 2004.

Please send bug reports, suggestions and other comments about GUAVA to support@gap-system.org. Currently known bugs and suggested GUAVA projects are listed on the bugs and projects web page http://cadigweb.ew.usna.edu/~wdj/gap/GUAVA/guava2do.html. Older releases and further history can be found on the GUAVA web page http://cadigweb.ew.usna.edu/~wdj/gap/GUAVA/.

Contributors: Other than the authors listed on the title page, the following people have contributed code to the GUAVA 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 GUAVA.

### Contents

1. Introduction
1.1 Introduction to the GUAVA package
1.2 Installing GUAVA
1.3 Loading GUAVA
2. Coding theory functions in GAP
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
4.7 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 WeightDistribution
4.9-2 InnerDistribution
4.9-3 DistancesDistribution
4.9-4 OuterDistribution
4.10 Decoding Functions
4.10-1 Decode
4.10-2 Decodeword
4.10-3 GeneralizedReedSolomonDecoderGao
4.10-4 GeneralizedReedSolomonListDecoder
4.10-5 NearestNeighborGRSDecodewords
4.10-6 NearestNeighborDecodewords
4.10-7 Syndrome
4.10-8 SyndromeTable
4.10-9 StandardArray
4.10-10 PermutationDecode
4.10-11 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 CheckMatCode
5.2-3 HammingCode
5.2-4 ReedMullerCode
5.2-5 AlternantCode
5.2-6 GoppaCode
5.2-7 GeneralizedSrivastavaCode
5.2-8 SrivastavaCode
5.2-9 CordaroWagnerCode
5.2-10 RandomLinearCode
5.2-11 OptimalityCode
5.2-12 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 QQRCode
5.5-8 FireCode
5.5-9 WholeSpaceCode
5.5-10 NullCode
5.5-11 RepetitionCode
5.5-12 CyclicCodes
5.5-13 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 GUAVA
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 CoefficientToPolynomial
7.6-7 DegreesMonomialTerm
7.6-8 DivisorsMultivariatePolynomial

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