guava - Gap

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GUAVA 10 7.3.3 SylvesterMat . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139 7.3.4 HadamardMat . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140 7.3.5 VandermondeMat . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140 7.3.6 PutStandardForm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141 7.3.7 IsInStandardForm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142 7.3.8 PermutedCols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142 7.3.9 VerticalConversionFieldMat . . . . . . . . . . . . . . . . . . . . . . . . . . 142 7.3.10 HorizontalConversionFieldMat . . . . . . . . . . . . . . . . . . . . . . . . 143 7.3.11 MOLS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143 7.3.12 IsLatinSquare . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144 7.3.13 AreMOLS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144 7.4 Some functions related to the norm of a code . . . . . . . . . . . . . . . . . . . . . 145 7.4.1 CoordinateNorm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145 7.4.2 CodeNorm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145 7.4.3 IsCoordinateAcceptable . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145 7.4.4 GeneralizedCodeNorm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146 7.4.5 IsNormalCode . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146 7.5 Miscellaneous functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146 7.5.1 CodeWeightEnumerator . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146 7.5.2 CodeDistanceEnumerator . . . . . . . . . . . . . . . . . . . . . . . . . . . 147 7.5.3 CodeMacWilliamsTransform . . . . . . . . . . . . . . . . . . . . . . . . . . 147 7.5.4 CodeDensity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147 7.5.5 SphereContent . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148 7.5.6 Krawtchouk . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148 7.5.7 PrimitiveUnityRoot . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148 7.5.8 PrimitivePolynomialsNr . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149 7.5.9 IrreduciblePolynomialsNr . . . . . . . . . . . . . . . . . . . . . . . . . . . 149 7.5.10 MatrixRepresentationOfElement . . . . . . . . . . . . . . . . . . . . . . . . 149 7.5.11 ReciprocalPolynomial . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150 7.5.12 CyclotomicCosets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150 7.5.13 WeightHistogram . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151 7.5.14 MultiplicityInList . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151 7.5.15 MostCommonInList . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151 7.5.16 RotateList . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152 7.5.17 CirculantMatrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152 7.6 Miscellaneous polynomial functions . . . . . . . . . . . . . . . . . . . . . . . . . . 152 7.6.1 MatrixTransformationOnMultivariatePolynomial . . . . . . . . . . . . . . . 152 7.6.2 DegreeMultivariatePolynomial . . . . . . . . . . . . . . . . . . . . . . . . . 152 7.6.3 DegreesMultivariatePolynomial . . . . . . . . . . . . . . . . . . . . . . . . 153 7.6.4 CoefficientMultivariatePolynomial . . . . . . . . . . . . . . . . . . . . . . . 153 7.6.5 SolveLinearSystem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154 7.6.6 GuavaVersion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154 7.6.7 ZechLog . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154 7.6.8 CoefficientToPolynomial . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155 7.6.9 DegreesMonomialTerm . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155 7.6.10 DivisorsMultivariatePolynomial . . . . . . . . . . . . . . . . . . . . . . . . 156
GUAVA 11 8 Coding theory functions in GAP 157 8.1 Distance functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157 8.1.1 AClosestVectorCombinationsMatFFEVecFFE . . . . . . . . . . . . . . . . . 157 8.1.2 AClosestVectorComb..MatFFEVecFFECoords . . . . . . . . . . . . . . . . 158 8.1.3 DistancesDistributionMatFFEVecFFE . . . . . . . . . . . . . . . . . . . . . 158 8.1.4 DistancesDistributionVecFFEsVecFFE . . . . . . . . . . . . . . . . . . . . 159 8.1.5 WeightVecFFE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159 8.1.6 DistanceVecFFE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159 8.2 Other functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160 8.2.1 ConwayPolynomial . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160 8.2.2 RandomPrimitivePolynomial . . . . . . . . . . . . . . . . . . . . . . . . . . 161 9 GNU Free Documentation License 162
Page 1 and 2: GUAVA A GAP4 Package for computing
Page 3 and 4: GUAVA 3 Contributors: Other than th
Page 5 and 6: GUAVA 5 4.2.1 + . . . . . . . . . .
Page 7 and 8: GUAVA 7 5.2.12 RandomLinearCode . .
Page 9: GUAVA 9 6.2.3 DirectProductCode . .
Page 13 and 14: GUAVA 13 a UNIX environment, where
Page 15 and 16: GUAVA 15 5 gap> Weight(c2); 9 gap>
Page 17 and 18: Chapter 3 Codewords Let GF(q) denot
Page 19 and 20: GUAVA 19 gap> VectorCodeword( c );
Page 21 and 22: GUAVA 21 gap> C:=HammingCode(3); a
Page 23 and 24: GUAVA 23 PolyCodeword returns a pol
Page 25 and 26: GUAVA 25 Example gap> a := Codeword
Page 27 and 28: GUAVA 27 Example gap> G := Generato
Page 29 and 30: GUAVA 29 GUAVA, unless one of the c
Page 31 and 32: GUAVA 31 4.3.2 IsSubset ♦ IsSubse
Page 33 and 34: GUAVA 33 Well-known MDS codes inclu
Page 35 and 36: GUAVA 35 gap> C:=ExtendedCode(C); a
Page 37 and 38: GUAVA 37 If the two codes C1 and C2
Page 39 and 40: GUAVA 39 4.5.3 LeftActingDomain ♦
Page 41 and 42: GUAVA 41 4.6.3 Display ♦ Display(
Page 43 and 44: GUAVA 43 [ 0*Z(5), 0*Z(5), 0*Z(5),
Page 45 and 46: GUAVA 45 MinimumDistance returns th
Page 47 and 48: GUAVA 47 the computation will finis
Page 49 and 50: GUAVA 49 36 4.8.6 DecreaseMinimumDi
Page 51 and 52: GUAVA 51 gap> a:=MinimumDistanceRan
Page 53 and 54: GUAVA 53 4.9.2 WeightDistribution
Page 55 and 56: GUAVA 55 C!.SpecialDecoder := funct
Page 57 and 58: GUAVA 57 GeneralizedReedSolomonList
Page 59 and 60: GUAVA 59 4.10.7 NearestNeighborDeco
Page 61 and 62:
GUAVA 61 [ [ 0 1 0 ], [ 1 0 1 ] ],
Page 63 and 64:
GUAVA 63 gap> MinimumDistance( C );
Page 65 and 66:
GUAVA 65 5.1.5 RandomCode ♦ Rando
Page 67 and 68:
GUAVA 67 elements of the code. The
Page 69 and 70:
GUAVA 69 5.2.7 GoppaCode ♦ GoppaC
Page 71 and 72:
GUAVA 71 (G,+) group with kernel K
Page 73 and 74:
GUAVA 73 . . . . . 1 . . . . . . .
Page 75 and 76:
GUAVA 75 Example gap> GabidulinCode
Page 77 and 78:
GUAVA 77 than n. A cyclic code is a
Page 79 and 80:
GUAVA 79 true gap> C3 := RootsCode(
Page 81 and 82:
GUAVA 81 true gap> IsCyclicCode(C1)
Page 83 and 84:
GUAVA 83 5.5.12 NullCode ♦ NullCo
Page 85 and 86:
GUAVA 85 gap> # gap> # This example
Page 87 and 88:
GUAVA 87 gap> IsSelfDualCode(C); tr
Page 89 and 90:
GUAVA 89 GeneralizedReedMullerCode(
Page 91 and 92:
GUAVA 91 5.7.2 AffinePointsOnCurve
Page 93 and 94:
GUAVA 93 • the support, • the c
Page 95 and 96:
GUAVA 95 support := [ Z(11)ˆ2, Z(1
Page 97 and 98:
GUAVA 97 5.7.15 RiemannRochSpaceBas
Page 99 and 100:
GUAVA 99 gap> A:=Z(11)ˆ0*[[1,2],[1
Page 101 and 102:
GUAVA 101 5.7.22 GoppaCodeClassical
Page 103 and 104:
GUAVA 103 This command returns a
Page 105 and 106:
GUAVA 105 gap> MinimumWeight(C); [2
Page 107 and 108:
GUAVA 107 gap> C2 := ExtendedCode(
Page 109 and 110:
GUAVA 109 Example gap> C1 := Hammin
Page 111 and 112:
GUAVA 111 the code, after which the
Page 113 and 114:
GUAVA 113 of C where a codeword of
Page 115 and 116:
GUAVA 115 Example gap> H := Hamming
Page 117 and 118:
GUAVA 117 6.2 Functions that Genera
Page 119 and 120:
GUAVA 119 gap> U := UnionCode(G, H)
Page 121 and 122:
GUAVA 121 of j − 1 auxiliary line
Page 123 and 124:
GUAVA 123 6.2.12 BZCodeNC ♦ BZCod
Page 125 and 126:
GUAVA 125 7.1.1 UpperBoundSingleton
Page 127 and 128:
GUAVA 127 7.1.6 UpperBoundGriesmer
Page 129 and 130:
GUAVA 129 gap> LowerBoundSpherePack
Page 131 and 132:
GUAVA 131 If the algorithm finds a
Page 133 and 134:
GUAVA 133 3 7.2.6 LowerBoundCoverin
Page 135 and 136:
GUAVA 135 where and This bound only
Page 137 and 138:
GUAVA 137 7.2.13 UpperBoundCovering
Page 139 and 140:
GUAVA 139 7.3.1 KrawtchoukMat ♦ K
Page 141 and 142:
GUAVA 141 [ Z(5)ˆ0, Z(5)ˆ0, Z(5)
Page 143 and 144:
GUAVA 143 Example gap> M := Z(9)*[[
Page 145 and 146:
GUAVA 145 Example gap> M := MOLS(4,
Page 147 and 148:
GUAVA 147 where A i is the number o
Page 149 and 150:
GUAVA 149 PrimitiveUnityRoot return
Page 151 and 152:
GUAVA 151 7.5.13 WeightHistogram
Page 153 and 154:
GUAVA 153 gap> poly:=yˆ2-x*(xˆ2-1
Page 155 and 156:
GUAVA 155 Returns the Zech log of x
Page 157 and 158:
Chapter 8 Coding theory functions i
Page 159 and 160:
GUAVA 159 Example gap> v:=[ Z(3)ˆ0
Page 161 and 162:
GUAVA 161 8.2.2 RandomPrimitivePoly
Page 163 and 164:
GUAVA 163 them. The ”Invariant Se
Page 165 and 166:
GUAVA 165 I. Preserve the section E
Page 167 and 168:
GUAVA 167 Each version of the Licen
Page 169 and 170:
GUAVA 169 [JH04] [Joy04] [Leo82] J.
Page 171 and 172:
GUAVA 171 code, Hadamard, 63 code,
Page 173 and 174:
GUAVA 173 linear code, 17 LowerBoun
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guava - Gap

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