Introduction

 
Construction of Additive Codes

      Construction of General Additive Codes
            AdditiveCode<F, K, n | L> : FldFin, RngIntElt, List -> Code
            AdditiveCode(G) : ModMatRngElt -> CodeAdd
            Example CodeAdd_AddLinDiff (H170E1)
            Example CodeAdd_CodeAddFromMatrix (H170E2)
            AdditiveCode(K, C) : FldFin, Code -> CodeAdd
            Example CodeAdd_CodeAddFromCode (H170E3)
            Example CodeAdd_CodeAddFromCodeFail (H170E4)

      Some Trivial Additive Codes
            AdditiveZeroCode(F, K, n) : FldFin, FldFin, RngIntElt -> CodeAdd
            AdditiveRepetitionCode(F, K, n) : FldFin, FldFin, RngIntElt -> Code
            AdditiveZeroSumCode(F, K, n) : FldFin, FldFin, RngIntElt -> Code
            AdditiveUniverseCode(F, K, n) : FldFin,FldFin, RngIntElt -> Code
            RandomAdditiveCode(F, K, n, k) : FldFin,FldFin, RngIntElt, RngIntElt -> Code
            Example CodeAdd_SimpleCodeChain (H170E5)

 
Invariants of an Additive Code

      The Ambient Space and Alphabet
            Alphabet(C) : Code -> Rng
            CoefficientField(C) : Code -> Rng
            AmbientSpace(C) : Code -> ModTupRng
            Generic(C) : Code -> Code
            Example CodeAdd_alphabet-coeff-field (H170E6)

      Basic Numerical Invariants
            Length(C) : Code -> RngIntElt
            Dimension(C) : Code -> FldRatElt
            NumberOfGenerators(C) : Code -> RngIntElt
            # C : Code -> RngIntElt
            InformationRate(C) : Code -> FldPrElt

      The Code Space
            GeneratorMatrix(C) : Code -> ModMatFldElt
            Basis(C) : Code -> [ ModTupRngElt ]
            C . i : Code, RngIntElt -> ModTupFldElt

      The Dual Space
            Dual(C) : CodeAdd -> CodeAdd
            ParityCheckMatrix(C) : Code -> ModMatFldElt

 
Operations on Codewords

      Construction of a Codeword
            C ! [a₁, ..., a_n] : Code, [ RngElt ] -> ModTupRngElt
            C ! u : Code, ModTupRngElt -> ModTupRngElt
            C ! 0 : Code, RngIntElt -> ModTupRngElt
            Random(C): Code -> ModTupRngElt

      Arithmetic Operations on Codewords
            u + v : ModTupRngElt, ModTupRngElt -> ModTupRngElt
            - u : ModTupRngElt -> ModTupRngElt
            u - v : ModTupRngElt, ModTupRngElt -> ModTupRngElt
            a * u : RngElt, ModTupRngElt -> ModTupRngElt
            Normalize(u) : ModTupRngElt -> ModTupRngElt

      Distance and Weight
            Distance(u, v) : ModTupRngElt, ModTupRngElt -> RngIntElt
            Weight(u) : ModTupRngElt -> RngIntElt

      Vector Space and Related Operations
            (u, v) : ModTupRngElt, ModTupRngElt -> RngElt
            TraceInnerProduct(K, u, v) : FldFin, ModTupFldElt, ModTupFldElt -> FldFinElt
            Support(w) : ModTupRngElt -> { RngIntElt }
            Coordinates(C, u) : Code, ModTupRngElt -> [ RngFinElt ]
            Parent(w): ModTupRngElt -> ModTupRng
            Rotate(u, k) : ModTupRngElt, RngIntElt -> ModTupRngElt
            Rotate(~u, k) : ModTupRngElt, RngIntElt ->
            Trace(u, S) : ModTupFldElt, FldFin -> ModTupFldElt

      Predicates for Codewords
            u eq v : ModTupRngElt, ModTupRngElt -> BoolElt
            u ne v : ModTupRngElt, ModTupRngElt -> BoolElt
            IsZero(u) : ModTupRngElt -> BoolElt

      Accessing Components of a Codeword
            u[i] : ModTupRngElt, RngIntElt -> RngElt
            u[i] := x;

 
Subcodes

      The Subcode Constructor
            sub<C | L> : CodeAdd, List -> CodeAdd
            Subcode(C, k) : CodeAdd, RngIntElt -> CodeAdd
            Subcode(C, S) : CodeAdd, { RngIntElt } -> Code
            SubcodeBetweenCode(C1, C2, k) : CodeAdd, CodeAdd, RngIntElt -> CodeAdd
            SubcodeWordsOfWeight(C, w) : CodeAdd, RngIntElt -> CodeAdd
            SubcodeWordsOfWeight(C, S) : CodeAdd, { RngIntElt } -> CodeAdd
            Example CodeAdd_SubcodeBetweenCode (H170E7)

      Sum, Intersection and Dual
            C + D : CodeAdd, CodeAdd -> CodeAdd
            C meet D : Code, Code -> Code
            Dual(C) : Code -> Code

      Membership and Equality
            u in C : ModTupRngElt, CodeAdd -> BoolElt
            u notin C : ModTupRngElt, CodeAdd -> BoolElt
            C subset D : Code, Code -> BoolElt
            C notsubset D : Code, Code -> BoolElt
            C eq D : Code, Code -> BoolElt
            C ne D : Code, Code -> BoolElt

 
Properties of Codes
      IsSelfDual(C) : Code -> BoolElt
      IsSelfOrthogonal(C) : Code -> BoolElt
      IsPerfect(C) : Code -> BoolElt
      IsProjective(C) : Code -> BoolElt
      IsAdditiveProjective(C) : CodeAdd -> BoolElt

 
The Weight Distribution

      The Minimum Weight
            MinimumWeight(C: parameters) : CodeAdd -> RngIntElt
            Example CodeAdd_additive-minweight (H170E8)

      The Weight Distribution
            WeightDistribution(C) : CodeAdd -> [ <RngIntElt, RngIntElt> ]
            DualWeightDistribution(C) : CodeAdd -> [ <RngIntElt, RngIntElt> ]

      The Weight Enumerator
            WeightEnumerator(C): CodeAdd -> RngMPolElt
            CompleteWeightEnumerator(C): CodeAdd -> RngMPolElt
            CompleteWeightEnumerator(C, u): Code, ModTupFldElt -> RngMPolElt

      The MacWilliams Transform
            MacWilliamsTransform(n, k, q, W) : RngIntElt, RngIntElt, RngIntElt, [ <RngIntElt, RngIntElt> ] -> [ <RngIntElt, RngIntElt> ]

      Words
            Words(C, w: parameters) : Code, RngIntElt -> { ModTupFldElt }
            NumberOfWords(C, w) : Code, RngIntElt -> RngIntElt
            WordsOfBoundedWeight(C, l, u: parameters) : Code, RngIntElt, RngIntElt -> { ModTupFldElt }

 
Families of Linear Codes

      Cyclic Codes
            AdditiveCyclicCode(v) : ModTupFldElt -> CodeAdd
            AdditiveCyclicCode(n, f) : RngIntElt, RngUPolElt -> CodeAdd
            AdditiveCyclicCode(v4, v2) : ModTupFldElt, ModTupFldElt -> CodeAdd
            AdditiveCyclicCode(n, f4, f2) : RngIntElt, RngUPolElt, RngUPolElt -> CodeAdd

      Quasicyclic Codes
            AdditiveQuasiCyclicCode(n, Q) : RngIntElt, SeqEnum[RngUPolElt] -> CodeAdd
            AdditiveQuasiCyclicCode(n, Q, h) : RngIntElt, SeqEnum[RngUPolElt], RngIntElt -> CodeAdd
            AdditiveQuasiCyclicCode(Q) : SeqEnum[ModTupFldElt] -> CodeAdd
            AdditiveQuasiCyclicCode(Q, h) : SeqEnum[ModTupFldElt], RngIntElt -> CodeAdd

 
New Codes from Old

      Standard Constructions
            AugmentCode(C) : CodeAdd -> Code
            CodeComplement(C, S) : Code, Code -> Code
            DirectSum(C, D) : Code, Code -> Code
            DirectSum(Q) : [Code] -> Code
            DirectProduct(C, D) : Code, Code -> Code
            ExtendCode(C) : CodeAdd -> CodeAdd
            ExtendCode(C, n) : Code, RngIntElt -> Code
            PadCode(C, n) : CodeAdd, RngIntElt -> CodeAdd
            PlotkinSum(C1, C2) : Code, Code -> Code
            PlotkinSum(C1, C2, C3: parameters) : Code,Code,Code -> Code
            PunctureCode(C, i) : CodeAdd, RngIntElt -> Code
            PunctureCode(C, S) : CodeAdd, { RngIntElt } -> Code
            ShortenCode(C, i) : Code, RngIntElt -> Code
            ShortenCode(C, S) : Code, { RngIntElt } -> Code

      Combining Codes
            C1 cat C2 : CodeAdd, CodeAdd -> CodeAdd
            Juxtaposition(C1, C2) : Code, Code -> Code

 
Automorphism Group
      AutomorphismGroup(C) : CodeAdd -> GrpPerm
      PermutationGroup(C) : CodeAdd -> GrpPerm

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