Introduction

 
Constructions for Z₄ Codes

      The Gray Map
            GrayMap(C) : Code -> Map
            GrayMapImage(C) : Code -> [ ModTupRngElt ]
            HasLinearGrayMapImage(C) : Code -> BoolElt, Code
            Example CodeZ4_GrayMap (H169E1)

      Families of Codes over Z₄
            KerdockCode(m): RngIntElt -> Code
            PreparataCode(m): RngIntElt -> Code
            ReedMullerCodeZ4(r, m) : RngIntElt, RngIntElt -> Code
            GoethalsCode(m) : RngIntElt -> Code
            DelsarteGoethalsCode(m, delta) : RngIntElt, RngIntElt -> Code
            GoethalsDelsarteCode(m, delta) : RngIntElt, RngIntElt -> Code
            QRCodeZ4(p) : RngIntElt -> Code
            GolayCodeZ4(e) : BoolElt -> Code
            SimplexAlphaCodeZ4(k) : RngIntElt -> Code
            SimplexBetaCodeZ4(k) : RngIntElt -> Code
            Example CodeZ4_Kerdock (H169E2)
            HadamardCodeZ4(δ, m) : RngIntElt, RngIntElt -> CodeLinRng, Mtrx
            ExtendedPerfectCodeZ4(δ, m) : RngIntElt, RngIntElt -> CodeLinRng, Mtrx
            Example CodeZ4_spain-Z4-1 (H169E3)
            ReedMullerCodeZ4(r, m) : RngIntElt, RngIntElt -> CodeLinRng
            ReedMullerCodesLRMZ4(r, m) : RngIntElt, RngIntElt -> SeqEnum
            ReedMullerCodeRMZ4(s, r, m) : RngIntElt, RngIntElt, RngIntElt -> CodeLinRng, Mtrx
            Example CodeZ4_spain-Z4-2 (H169E4)
            ReedMullerCodesRMZ4(s, m) : RngIntElt, RngIntElt -> Tup
            Example CodeZ4_spain-Z4-3 (H169E5)

      Derived Binary Codes
            BinaryResidueCode(C) : Code -> Code
            BinaryTorsionCode(C) : Code -> Code
            Z4CodeFromBinaryChain(C1, C2) : Code, Code -> Code
            Example CodeZ4_derived-binary (H169E6)

      New Codes from Old
            PlotkinSum(A, B) : Mtrx, Mtrx -> Mtrx
            PlotkinSum(C, D) : Code, Code -> Code
            QuaternaryPlotkinSum(A, B) : Mtrx, Mtrx -> Mtrx
            QuaternaryPlotkinSum(C, D) : Code, Code -> Code
            BQPlotkinSum(A, B, C) : Mtrx, Mtrx, Mtrx -> Mtrx
            BQPlotkinSum(D, E, F) : Code, Code, Code -> Code
            DoublePlotkinSum(A, B, C, D) : Mtrx, Mtrx, Mtrx, Mtrx -> Mtrx
            DoublePlotkinSum(E, F, G, H) : Code, Code, Code, Code -> Code
            DualKroneckerZ4(C) : CodeLinRng -> CodeLinRng
            Example CodeZ4_spain-Z4-4 (H169E7)
            Example CodeZ4_spain-Z4-4a (H169E8)

 
Invariants

      The Standard Form
            StandardForm(C) : Code -> Code, Map
            Example CodeZ4_StandardForm (H169E9)
            StandardFormDual(C) : CodeLinRng -> CodeLinRng, Map

      Structures Associated with the Gray Map
            MinRowsGeneratorMatrix(C) : CodeLinRng -> ModMatRngElt
            MinRowsParityCheckMatrix(C) : CodeLinRng -> ModMatRngElt
            DualZ4(C) : CodeLinRng -> CodeLinRng
            Example CodeZ4_new-invar (H169E10)
            SpanZ2CodeZ4(C) : CodeLinRng -> CodeLinFld
            KernelZ2CodeZ4(C) : CodeLinRng -> CodeLinRng
            KernelCosetRepresentatives(C) : CodeLinRng -> SeqEnum, SeqEnum
            DimensionOfSpanZ2(C) : CodeLinRng -> RngIntElt
            DimensionOfKernelZ2(C) : CodeLinRng -> RngIntElt
            Example CodeZ4_spain-Z4-5 (H169E11)

      Coset Representatives
            CosetRepresentatives(C) : CodeLinRng -> SeqEnum
            CosetRepresentatives(C, S) : CodeLinRng, CodeLinRng -> SeqEnum, SeqEnum
            Example CodeZ4_spain-Z4-7 (H169E12)

      Information Space and Information Sets
            InformationSpace(C) : CodeLinRng -> ModTupRng, ModTupFld, Map, Map
            Example CodeZ4_spain-Z4-9 (H169E13)
            InformationSet(C) : CodeLinRng -> ModTupRng, ModTupFld, Map, Map
            IsInformationSet(C, I) : CodeLinRng, [RngIntElt] -> BoolElt, BoolElt
            Example CodeZ4_spain-Z4-10 (H169E14)

      Syndrome Space and Coset Leaders
            SyndromeSpace(C) : CodeLinRng -> ModTupRng, ModTupFld
            Syndrome(u, C) : ModTupFldElt, CodeLinRng -> ModTupRngElt
            CosetLeaders(C) : CodeLinRng -> SetIndx, Map
            Example CodeZ4_spain-Z4-11 (H169E15)

      Miscellaneous Functions
            Correlation(v) : ModTupRngElt -> RngQuadElt

 
Weight Distributions

      Hamming Weight
            MinimumWeight(C) : Code -> RngIntElt
            WeightDistribution(C) : Code -> [ <RngIntElt, RngIntElt> ]
            DualWeightDistribution(C) : Code -> [ <RngIntElt, RngIntElt> ]
            Example CodeZ4_weight-dist-k8 (H169E16)

      Lee Weight
            LeeWeight(a) : RngIntRes -> RngIntElt
            LeeWeight(v) : ModTupRngElt -> RngIntElt
            LeeDistance(u, v) : ModTupRngElt, ModTupRngElt -> RngIntElt
            MinimumLeeWeight(C) : Code -> RngIntElt
            LeeWeightDistribution(C) : Code -> SeqEnum
            DualLeeWeightDistribution(C) : Code -> SeqEnum
            WordsOfLeeWeight(C, w) : Code, RngIntElt -> SetEnum
            WordsOfBoundedLeeWeight(C, l, u) : Code, RngIntElt, RngIntElt -> SetEnum
            Example CodeZ4_lee-dist (H169E17)

      Euclidean Weight
            EuclideanWeight(a) : RngIntRes -> RngIntElt
            EuclideanWeight(v) : ModTupRngElt -> RngIntElt
            EuclideanDistance(u, v) : ModTupRngElt, ModTupRngElt -> RngIntElt
            MinimumEuclideanWeight(C) : Code -> RngIntElt
            EuclideanWeightDistribution(C) : Code -> SeqEnum
            DualEuclideanWeightDistribution(C) : Code -> SeqEnum
            Example CodeZ4_euclidean-dist (H169E18)

 
Weight Enumerators
      CompleteWeightEnumerator(C): Code -> RngMPolElt
      SymmetricWeightEnumerator(C): Code -> RngMPolElt
      WeightEnumerator(C): Code -> RngMPolElt
      LeeWeightEnumerator(C): Code -> RngMPolElt
      EuclideanWeightEnumerator(C): Code -> RngMPolElt
      Example CodeZ4_WeightEnumerator (H169E19)

 
Decoding

      Coset Decoding
            CosetDecode(C, u : parameters) : CodeLinRng, ModTupFldElt -> BoolElt, ModTupRngElt, ModTupFldElt
            CosetDecode(C, Q : parameters) : CodeLinRng, [ModTupFldElt] -> SeqEnum, SeqEnum, SeqEnum
            Example CodeZ4_spain-Z4-12 (H169E20)

      Syndrome Decoding
            SyndromeDecode(C, u) : CodeLinRng, ModTupFldElt -> BoolElt, ModTupRngElt, ModTupFldElt
            SyndromeDecode(C, Q) : CodeLinRng, [ModTupFldElt] -> SeqEnum, SeqEnum, SeqEnum
            Example CodeZ4_spain-Z4-13 (H169E21)

      Lifted Decoding
            LiftedDecode(C, u : parameters) : CodeLinRng, ModTupFldElt -> BoolElt, ModTupRngElt, ModTupFldElt
            LiftedDecode(C, Q : parameters) : CodeLinRng, [ModTupFldElt] -> SeqEnum, SeqEnum, SeqEnum
            Example CodeZ4_hadamard_decoding (H169E22)

      Permutation Decoding
            IsPermutationDecodeSet(C, I, S, s) : CodeLinRng, [RngIntElt], [GrpPermElt], RngIntElt -> BoolElt
            PermutationDecode(C, I, S, s, u) : CodeLinRng, [RngIntElt], [GrpPermElt], RngIntElt, ModTupFldElt -> BoolElt, ModTupRngElt, ModTupFldElt
            PermutationDecode(C, I, S, s, Q) : CodeLinRng, [RngIntElt], [GrpPermElt], RngIntElt, [ModTupFldElt] -> [BoolElt], [ModTupRngElt], [ModTupFldElt]
            Example CodeZ4_spain-Z4-15 (H169E23)
            PDSetHadamardCodeZ4(δ, m) : RngIntElt, RngIntElt -> SeqEnum, SeqEnum, SeqEnum, SeqEnum, SeqEnum
            PDSetKerdockCodeZ4(m) : RngIntElt -> SeqEnum, SeqEnum, SeqEnum, SeqEnum
            Example CodeZ4_spain-Z4-16 (H169E24)

 
Automorphism Groups
      PermutationGroupHadamardCodeZ4(δ, m) : RngIntElt, RngIntElt -> GrpPerm, Mtrx
      PermutationGroupHadamardCodeZ4Order(δ, m) : RngIntElt, RngIntElt -> RngIntElt
      PermutationGroupExtendedPerfectCodeZ4(δ, m) : RngIntElt, RngIntElt -> GrpPerm, Mtrx
      PermutationGroupExtendedPerfectCodeZ4Order(δ, m) : RngIntElt, RngIntElt -> RngIntElt
      Example CodeZ4_spain-Z4-8 (H169E25)
      PermutationGroup(C) : CodeLinRng -> GrpPerm
      PermutationGroupGrayMapImage(C) : CodeLinRng -> GrpPerm
      Example CodeZ4_code-z4-perm-group (H169E26)

 
Bibliography

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