guava - Gap

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GUAVA 42 4.7 Generating (Check) Matrices and Polynomials 4.7.1 GeneratorMat ♦ GeneratorMat(C) (function) GeneratorMat returns a generator matrix of C. The code consists of all linear combinations of the rows of this matrix. If until now no generator matrix of C was determined, it is computed from either the parity check matrix, the generator polynomial, the check polynomial or the elements (if possible), whichever is available. If C is a non-linear code, the function returns an error. Example gap> GeneratorMat( HammingCode( 3, GF(2) ) ); [ [ an immutable GF2 vector of length 7], [ an immutable GF2 vector of length 7], [ an immutable GF2 vector of length 7], [ an immutable GF2 vector of length 7] ] gap> Display(last); 1 1 1 . . . . 1 . . 1 1 . . . 1 . 1 . 1 . 1 1 . 1 . . 1 gap> GeneratorMat( RepetitionCode( 5, GF(25) ) ); [ [ Z(5)ˆ0, Z(5)ˆ0, Z(5)ˆ0, Z(5)ˆ0, Z(5)ˆ0 ] ] gap> GeneratorMat( NullCode( 14, GF(4) ) ); [ ] 4.7.2 CheckMat ♦ CheckMat(C) (function) CheckMat returns a parity check matrix of C. The code consists of all words orthogonal to each of the rows of this matrix. The transpose of the matrix is a right inverse of the generator matrix. The parity check matrix is computed from either the generator matrix, the generator polynomial, the check polynomial or the elements of C (if possible), whichever is available. If C is a non-linear code, the function returns an error. Example gap> CheckMat( HammingCode(3, GF(2) ) ); [ [ 0*Z(2), 0*Z(2), 0*Z(2), Z(2)ˆ0, Z(2)ˆ0, Z(2)ˆ0, Z(2)ˆ0 ], [ 0*Z(2), Z(2)ˆ0, Z(2)ˆ0, 0*Z(2), 0*Z(2), Z(2)ˆ0, Z(2)ˆ0 ], [ Z(2)ˆ0, 0*Z(2), Z(2)ˆ0, 0*Z(2), Z(2)ˆ0, 0*Z(2), Z(2)ˆ0 ] ] gap> Display(last); . . . 1 1 1 1 . 1 1 . . 1 1 1 . 1 . 1 . 1 gap> CheckMat( RepetitionCode( 5, GF(25) ) ); [ [ Z(5)ˆ0, Z(5)ˆ2, 0*Z(5), 0*Z(5), 0*Z(5) ], [ 0*Z(5), Z(5)ˆ0, Z(5)ˆ2, 0*Z(5), 0*Z(5) ], [ 0*Z(5), 0*Z(5), Z(5)ˆ0, Z(5)ˆ2, 0*Z(5) ],
GUAVA 43 [ 0*Z(5), 0*Z(5), 0*Z(5), Z(5)ˆ0, Z(5)ˆ2 ] ] gap> CheckMat( WholeSpaceCode( 12, GF(4) ) ); [ ] 4.7.3 GeneratorPol ♦ GeneratorPol(C) (function) GeneratorPol returns the generator polynomial of C. The code consists of all multiples of the generator polynomial modulo x n − 1, where n is the word length of C. The generator polynomial is determined from either the check polynomial, the generator or check matrix or the elements of C (if possible), whichever is available. If C is not a cyclic code, the function returns ‘false’. Example gap> GeneratorPol(GeneratorMatCode([[1, 1, 0], [0, 1, 1]], GF(2))); Z(2)ˆ0+x_1 gap> GeneratorPol( WholeSpaceCode( 4, GF(2) ) ); Z(2)ˆ0 gap> GeneratorPol( NullCode( 7, GF(3) ) ); -Z(3)ˆ0+x_1ˆ7 4.7.4 CheckPol ♦ CheckPol(C) (function) CheckPol returns the check polynomial of C. The code consists of all polynomials f with f · h ≡ 0 (mod x n − 1), where h is the check polynomial, and n is the word length of C. The check polynomial is computed from the generator polynomial, the generator or parity check matrix or the elements of C (if possible), whichever is available. If C if not a cyclic code, the function returns an error. Example gap> CheckPol(GeneratorMatCode([[1, 1, 0], [0, 1, 1]], GF(2))); Z(2)ˆ0+x_1+x_1ˆ2 gap> CheckPol(WholeSpaceCode(4, GF(2))); Z(2)ˆ0+x_1ˆ4 gap> CheckPol(NullCode(7,GF(3))); Z(3)ˆ0 4.7.5 RootsOfCode ♦ RootsOfCode(C) (function) RootsOfCode returns a list of all zeros of the generator polynomial of a cyclic code C. These are finite field elements in the splitting field of the generator polynomial, GF(q m ), m is the multiplicative order of the size of the base field of the code, modulo the word length.
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 and 10: GUAVA 9 6.2.3 DirectProductCode . .
Page 11 and 12: GUAVA 11 8 Coding theory functions
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: GUAVA 41 4.6.3 Display ♦ Display(
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
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GUAVA 93 • the support, • the c
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GUAVA 95 support := [ Z(11)ˆ2, Z(1
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GUAVA 97 5.7.15 RiemannRochSpaceBas
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GUAVA 99 gap> A:=Z(11)ˆ0*[[1,2],[1
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GUAVA 101 5.7.22 GoppaCodeClassical
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GUAVA 103 This command returns a
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GUAVA 105 gap> MinimumWeight(C); [2
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GUAVA 107 gap> C2 := ExtendedCode(
Page 109 and 110:
GUAVA 109 Example gap> C1 := Hammin
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GUAVA 111 the code, after which the
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GUAVA 113 of C where a codeword of
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GUAVA 115 Example gap> H := Hamming
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GUAVA 117 6.2 Functions that Genera
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GUAVA 119 gap> U := UnionCode(G, H)
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GUAVA 121 of j − 1 auxiliary line
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GUAVA 123 6.2.12 BZCodeNC ♦ BZCod
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GUAVA 125 7.1.1 UpperBoundSingleton
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GUAVA 127 7.1.6 UpperBoundGriesmer
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GUAVA 129 gap> LowerBoundSpherePack
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GUAVA 131 If the algorithm finds a
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GUAVA 133 3 7.2.6 LowerBoundCoverin
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GUAVA 135 where and This bound only
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GUAVA 137 7.2.13 UpperBoundCovering
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GUAVA 139 7.3.1 KrawtchoukMat ♦ K
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GUAVA 141 [ Z(5)ˆ0, Z(5)ˆ0, Z(5)
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GUAVA 143 Example gap> M := Z(9)*[[
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GUAVA 145 Example gap> M := MOLS(4,
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GUAVA 147 where A i is the number o
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GUAVA 149 PrimitiveUnityRoot return
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GUAVA 151 7.5.13 WeightHistogram
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GUAVA 153 gap> poly:=yˆ2-x*(xˆ2-1
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GUAVA 155 Returns the Zech log of x
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Chapter 8 Coding theory functions i
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GUAVA 159 Example gap> v:=[ Z(3)ˆ0
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GUAVA 161 8.2.2 RandomPrimitivePoly
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GUAVA 163 them. The ”Invariant Se
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GUAVA 165 I. Preserve the section E
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GUAVA 167 Each version of the Licen
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GUAVA 169 [JH04] [Joy04] [Leo82] J.
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GUAVA 171 code, Hadamard, 63 code,
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GUAVA 173 linear code, 17 LowerBoun
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