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LINEAR CODES OVER FINITE FIELDS

 
Acknowledgements
 
Introduction
 
Construction of Codes
      Construction of General Linear Codes
      Some Trivial Linear Codes
      Some Basic Families of Codes
 
Invariants of a Code
      Basic Numerical Invariants
      The Ambient Space and Alphabet
      The Code Space
      The Dual Space
      The Information Space and Information Sets
      The Syndrome Space
      The Generator Polynomial
 
Operations on Codewords
      Construction of a Codeword
      Arithmetic Operations on Codewords
      Distance and Weight
      Vector Space and Related Operations
      Predicates for Codewords
      Accessing Components of a Codeword
 
Coset Leaders
 
Subcodes
      The Subcode Constructor
      Sum, Intersection and Dual
      Membership and Equality
 
Properties of Codes
 
The Weight Distribution
      The Minimum Weight
      The Weight Distribution
      The Weight Enumerator
      The MacWilliams Transform
      Words
      Covering Radius and Diameter
 
Families of Linear Codes
      Cyclic and Quasicyclic Codes
      BCH Codes and their Generalizations
      Quadratic Residue Codes and their Generalizations
      Reed--Solomon and Justesen Codes
      Maximum Distance Separable Codes
 
New Codes from Existing
      Standard Constructions
      Changing the Alphabet of a Code
      Combining Codes
 
Coding Theory and Cryptography
      Standard Attacks
      Generalized Attacks
 
Bounds
      Best Known Bounds for Linear Codes
      Bounds on the Cardinality of a Largest Code
      Bounds on the Minimum Distance
      Asymptotic Bounds on the Information Rate
      Other Bounds
 
Best Known Linear Codes
 
Decoding
 
Transforms
      Mattson--Solomon Transforms
      Krawchouk Polynomials
 
Automorphism Groups
      Introduction
      Group Actions
      Automorphism Group
      Equivalence and Isomorphism of Codes
 
Bibliography







DETAILS

 
Introduction

 
Construction of Codes

      Construction of General Linear Codes
            LinearCode<R, n | L> : FldFin, RngIntElt, List -> Code
            Example CodeFld_TernaryGolayCode (H152E1)
            LinearCode(U) : ModTupRng -> Code
            LinearCode(A) : ModMatRngElt -> Code
            Example CodeFld_CodeFromMatrix (H152E2)
            PermutationCode(u, G) : ModTupRngElt, GrpPerm -> Code
            Example CodeFld_PermutationCode (H152E3)

      Some Trivial Linear Codes
            ZeroCode(R, n) : FldFin, RngIntElt -> Code
            RepetitionCode(R, n) : FldFin, RngIntElt -> Code
            ZeroSumCode(R, n) : FldFin, RngIntElt -> Code
            UniverseCode(R, n) : FldFin, RngIntElt -> Code
            EvenWeightCode(n) : RngIntElt -> Code
            EvenWeightSubcode(C) : Code -> Code
            RandomLinearCode(K, n, k) : FldFin, RngIntElt, RngIntElt -> Code
            CordaroWagnerCode(n) : RngIntElt -> Code
            Example CodeFld_SimpleCodeChain (H152E4)

      Some Basic Families of Codes
            CyclicCode(n, g) : RngIntElt, RngUPolElt -> Code
            Example CodeFld_CyclicCode (H152E5)
            HammingCode(K, r) : FldFin, RngIntElt -> Code
            Example CodeFld_HammingCode (H152E6)
            SimplexCode(r) : RngIntElt -> Code
            ReedMullerCode(r, m) : RngIntElt, RngIntElt -> Code
            Example CodeFld_ReedMullerCode (H152E7)

 
Invariants of a Code

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

      The Ambient Space and Alphabet
            AmbientSpace(C) : Code -> ModTupRng
            RSpace(C) : Code -> ModTupRng
            Generic(C) : Code -> Code
            Alphabet(C) : Code -> Rng

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

      The Dual Space
            Dual(C) : Code -> Code
            ParityCheckMatrix(C) : Code -> ModMatFldElt
            Example CodeFld_GeneratorMatrix (H152E8)
            Hull(C) : Code -> Code

      The Information Space and Information Sets
            InformationSpace(C) : Code -> ModTupFld
            InformationSet(C) : Code -> [ RngIntElt ]
            AllInformationSets(C) : Code -> [ [ RngIntElt ] ]
            StandardForm(C) : Code -> Code, Map
            Example CodeFld_StandardForm (H152E9)

      The Syndrome Space
            SyndromeSpace(C) : Code -> ModTupFld

      The Generator Polynomial
            GeneratorPolynomial(C) : Code -> RngUPolElt
            CheckPolynomial(C) : Code -> RngUPolElt
            Idempotent(C) : Code -> RngUPolElt
            Example CodeFld_GeneratorPolynomial (H152E10)

 
Operations on Codewords

      Construction of a Codeword
            C ! [a1, ..., an] : 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
            Syndrome(w, C) : ModTupFldElt, Code -> ModTupFldElt

      Distance and Weight
            Distance(u, v) : ModTupRngElt, ModTupRngElt -> RngIntElt
            Weight(u) : ModTupRngElt -> RngIntElt
            LeeWeight(u) : ModTupRngElt -> RngIntElt
            Example CodeFld_Distance (H152E11)

      Vector Space and Related Operations
            (u, v) : ModTupRngElt, ModTupRngElt -> RngElt
            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
            Example CodeFld_Coordinates (H152E12)

      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;

 
Coset Leaders
      CosetLeaders(C) : Code -> {@ ModTupFldElt @}, Map
      Example CodeFld_CosetLeaders (H152E13)

 
Subcodes

      The Subcode Constructor
            sub<C | L> : Code, List -> Code
            Subcode(C, k) : Code, RngIntElt -> Code
            Subcode(C, S) : Code, { RngIntElt } -> Code
            SubcodeBetweenCode(C1, C2, k) : Code, Code, RngIntElt -> Code
            SubcodeWordsOfWeight(C, S) : Code, { RngIntElt } -> Code
            Example CodeFld_SubcodeBetweenCode (H152E14)

      Sum, Intersection and Dual
            C + D : Code, Code -> Code
            C meet D : Code, Code -> Code
            Dual(C) : Code -> Code
            Example CodeFld_SumIntersection (H152E15)
            Example CodeFld_DualRS (H152E16)

      Membership and Equality
            u in C : ModTupRngElt, Code -> BoolElt
            u notin C : ModTupRngElt, Code -> 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
      IsCyclic(C) : Code -> BoolElt
      IsSelfDual(C) : Code -> BoolElt
      IsSelfOrthogonal(C) : Code -> BoolElt
      IsMaximumDistanceSeparable(C) : Code -> BoolElt
      IsEquidistant(C) : Code -> BoolElt
      IsPerfect(C) : Code -> BoolElt
      IsNearlyPerfect(C) : Code -> BoolElt
      IsEven(C) : Code -> BoolElt
      IsDoublyEven(C) : Code -> BoolElt
      IsProjective(C) : Code -> BoolElt
      Example CodeFld_SelfDual (H152E17)
      Example CodeFld_SelfOrthogonal (H152E18)

 
The Weight Distribution

      The Minimum Weight
            MinimumWeight(C: parameters) : Code -> RngIntElt
            MinimumWeightBounds(C) : Code -> RngIntElt, RngIntElt
            ResetMinimumWeightBounds(C) : Code ->
            VerifyMinimumDistanceLowerBound(C, d) : Code, RngIntElt -> BoolElt, RngIntElt, BoolElt
            VerifyMinimumDistanceUpperBound(C, d) : Code, RngIntElt -> BoolElt, RngIntElt, BoolElt
            MinimumWord(C) : Code -> ModTupFldElt
            MinimumWords(C) : Code -> { ModTupFldElt }
            Example CodeFld_VerifyLower (H152E19)
            IncludeAutomorphism(~C, p) : Code, GrpPermElt ->
            KnownAutomorphismSubgroup(C) : Code -> GrpPerm

      The Weight Distribution
            WeightDistribution(C) : Code -> [ <RngIntElt, RngIntElt> ]
            WeightDistribution(C, u) : Code, ModTupFldElt -> [ <RngIntElt, RngIntElt> ]
            DualWeightDistribution(C) : Code -> [ <RngIntElt, RngIntElt> ]
            Example CodeFld_WeightDistribution (H152E20)
            PartialWeightDistribution(C, ub) : Code -> [ <RngIntElt, RngIntElt> ]

      The Weight Enumerator
            WeightEnumerator(C): Code -> RngMPolElt
            WeightEnumerator(C, u): Code, ModTupFldElt -> RngMPolElt
            CompleteWeightEnumerator(C): Code -> RngMPolElt
            CompleteWeightEnumerator(C, u): Code, ModTupFldElt -> RngMPolElt
            Example CodeFld_WeightEnumerator (H152E21)

      The MacWilliams Transform
            MacWilliamsTransform(n, k, q, W) : RngIntElt, RngIntElt, RngIntElt, [ <RngIntElt, RngIntElt> ] -> [ <RngIntElt, RngIntElt> ]
            MacWilliamsTransform(n, k, K, W) : RngIntElt, RngIntElt, FldFin, RngMPol -> RngMPol
            Example CodeFld_MacWilliams (H152E22)

      Words
            Words(C, w: parameters) : Code, RngIntElt -> { ModTupFldElt }
            NumberOfWords(C, w) : Code, RngIntElt -> RngIntElt
            WordsOfBoundedWeight(C, l, u: parameters) : Code, RngIntElt, RngIntElt -> { ModTupFldElt }
            ConstantWords(C, i) : Code, RngIntElt -> { ModTupFldElt }
            NumberOfConstantWords(C, i) : Code, RngIntElt -> RngIntElt
            Example CodeFld_Words (H152E23)

      Covering Radius and Diameter
            CosetDistanceDistribution(C) : Code -> [ <RngIntElt, RngIntElt> ]
            CoveringRadius(C) : Code -> RngIntElt
            Diameter(C) : Code -> RngIntElt
            Example CodeFld_CoveringRadius (H152E24)

 
Families of Linear Codes

      Cyclic and Quasicyclic Codes
            CyclicCode(u) : ModTupRngElt -> Code
            CyclicCode(n, T, K) : RngIntElt, [ FldFinElt ], FldFin -> Code
            QuasiCyclicCode(n, Gen) : RngIntElt, [ RngUPolElt ] -> Code
            QuasiCyclicCode(Gen) : [ ModTupRngElt ] -> Code
            QuasiCyclicCode(n, Gen, h) : RngIntElt, [ RngUPolElt ], RngIntElt -> Code
            QuasiCyclicCode(Gen, h) : [ModTupRngElt] , RngIntElt -> Code
            ConstaCyclicCode(n, f, alpha) : RngUPolElt, RngIntElt, FldFinElt -> Code
            QuasiTwistedCyclicCode(n, Gen, alpha) : RngIntElt, [RngUPolElt], FldFinElt -> Code
            QuasiTwistedCyclicCode(Gen, alpha) : [ModTupRngElt], FldFinElt -> Code
            Example CodeFld_ChainCyclic (H152E25)

      BCH Codes and their Generalizations
            BCHCode(K, n, d, b) : FldFin, RngIntElt, RngIntElt, RngIntElt -> Code
            Example CodeFld_BCHCode (H152E26)
            GoppaCode(L, G) : [ FldFinElt ], RngUPolElt -> Code
            Example CodeFld_GoppaCode (H152E27)
            ChienChoyCode(P, G, n, S) : RngUPolElt, RngUPolElt, RngIntElt, FldFin -> Code
            AlternantCode(A, Y, r, S) : [ FldFinElt ], [ FldFinElt ], RngIntElt, FldFin -> Code
            Example CodeFld_AlternantCode (H152E28)
            NonPrimitiveAlternantCode(n, m, r) : RngIntElt,RngIntElt,RngIntElt->Code
            FireCode(h, s, n) : RngUPolElt, RngIntElt, RngIntElt -> Code
            GabidulinCode(A, W, Z, t) : [ FldFinElt ], [ FldFinElt ], [ FldFinElt ], RngIntElt -> Code
            SrivastavaCode(A, W, mu, S) : [ FldFinElt ], [ FldFinElt ], RngIntElt, FldFin -> Code
            GeneralizedSrivastavaCode(A, W, Z, t, S) : [ FldFinElt ], [ FldFinElt ], [ FldFinElt ], RngIntElt, FldFin -> Code

      Quadratic Residue Codes and their Generalizations
            QRCode(K, n) : FldFin, RngIntElt -> Code
            GolayCode(K, ext) : FldFin, BoolElt -> Code
            DoublyCirculantQRCode(p) : RngIntElt -> Code
            DoublyCirculantQRCodeGF4(m, a) : RngIntElt, RngElt -> Code
            BorderedDoublyCirculantQRCode(p, a, b) : RngIntElt, RngElt, RngElt -> Code
            TwistedQRCode(l, m) : RngIntElt,RngIntElt -> Code
            PowerResidueCode(K, n, p) : FldFin, RngIntElt, RngIntElt -> Code
            Example CodeFld_QuadraticResidueCode (H152E29)

      Reed--Solomon and Justesen Codes
            ReedSolomonCode(K, d, b) : FldFin, RngIntElt, RngIntElt -> Code
            ReedSolomonCode(n, d) : RngIntElt, RngIntElt -> Code
            GRSCode(A, V, k) : [ FldFinElt ], [ FldFinElt ], RngIntElt -> Code
            JustesenCode(N, K) : Code, FldFinElt, RngIntElt -> Code
            Example CodeFld_GRSCode (H152E30)

      Maximum Distance Separable Codes
            MDSCode(K, k) : FldFin, RngIntElt -> Code

 
New Codes from Existing

      Standard Constructions
            AugmentCode(C) : Code -> Code
            CodeComplement(C, C1) : Code, Code -> Code
            DirectSum(C, D) : Code, Code -> Code
            DirectSum(Q) : [Code] -> Code
            DirectProduct(C, D) : Code, Code -> Code
            ExtendCode(C) : Code -> Code
            ExtendCode(C, n) : Code, RngIntElt -> Code
            PadCode(C, n) : Code, RngIntElt -> Code
            ExpurgateCode(C) : Code -> Code
            ExpurgateCode(C, L) : Code,[ModTupFldElt] -> Code
            ExpurgateWeightCode(C, w) : Code, RngIntElt -> Code
            LengthenCode(C) : Code -> Code
            PlotkinSum(C1, C2) : Code, Code -> Code
            PlotkinSum(C1, C2, C3: parameters) : Code, Code, Code -> Code
            PunctureCode(C, i) : Code, RngIntElt -> Code
            PunctureCode(C, S) : Code, { RngIntElt } -> Code
            ShortenCode(C, i) : Code, RngIntElt -> Code
            ShortenCode(C, S) : Code, { RngIntElt } -> Code
            Example CodeFld_Make12-8-4Code (H152E31)

      Changing the Alphabet of a Code
            ExtendField(C, L) : Code, FldFin -> Code, Map
            LinearCode(C, S) : Code, FldFin -> Code, Map
            SubfieldRepresentationCode(C, K) : Code, FldFin -> Code
            SubfieldRepresentationParityCode(C, K) : Code, FldFin -> Code
            SubfieldSubcode(C, S) : Code, FldFin -> Code, Map
            SubfieldCode(C, S) : Code, FldFin -> Code
            Trace(C, F) : Code, FldFin -> Code

      Combining Codes
            C1 cat C2 : Code, Code -> Code
            Juxtaposition(C1, C2) : Code, Code -> Code
            ConcatenatedCode(O, I) : Code, Code -> Code
            Example CodeFld_ConcatenatedCode (H152E32)
            ConstructionX(C1, C2, C3) : Code, Code, Code -> Code
            ConstructionXChain(S, C) : [Code], Code -> [Code]
            Example CodeFld_constructionX (H152E33)
            ConstructionX3(C1, C2, C3, D1, D2) : Code, Code, Code, Code, Code -> Code, Map
            ConstructionX3u(C1, C2, C3, D1, D2) : Code, Code, Code, Code, Code -> Code, Code
            Example CodeFld_X3 (H152E34)
            ConstructionXX(C1, C2, C3, D2, D3) : Code, Code, Code, Code, Code -> Code
            Example CodeFld_XX (H152E35)
            ZinovievCode(I, O) : [Code], [Code] -> Code
            Example CodeFld_Zinoviev (H152E36)
            ConstructionY1(C) : Code -> Code
            ConstructionY1(C, w) : Code, RngIntElt -> Code

 
Coding Theory and Cryptography

      Standard Attacks
            McEliecesAttack(C, v, e) : Code, ModTupFldElt, RngIntElt -> ModTupFldElt
            LeeBrickellsAttack(C, v, e, p) : Code, ModTupFldElt, RngIntElt, RngIntElt -> ModTupFldElt
            LeonsAttack(C, v, e, p, l) : Code, ModTupFldElt, RngIntElt, RngIntElt,RngIntElt -> ModTupFldElt
            SternsAttack(C, v, e, p, l) : Code, ModTupFldElt, RngIntElt, RngIntElt,RngIntElt -> ModTupFldElt
            CanteautChabaudsAttack(C, v, e, p, l) : Code, ModTupFldElt, RngIntElt, RngIntElt,RngIntElt -> ModTupFldElt

      Generalized Attacks
            DecodingAttack(C, v, e) : Code, ModTupFldElt, RngIntElt, RngIntElt,RngIntElt -> ModTupFldElt

 
Bounds

      Best Known Bounds for Linear Codes
            BKLCLowerBound(F, n, k) : FldFin, RngIntElt, RngIntElt -> RngIntElt
            BKLCUpperBound(F, n, k) : FldFin, RngIntElt, RngIntElt -> RngIntElt
            BLLCLowerBound(F, k, d) : FldFin, RngIntElt, RngIntElt -> RngIntElt
            BLLCUpperBound(F, k, d) : FldFin, RngIntElt, RngIntElt -> RngIntElt
            BDLCLowerBound(F, n, d) : FldFin, RngIntElt, RngIntElt -> RngIntElt
            BDLCUpperBound(F, n, d) : FldFin, RngIntElt, RngIntElt -> RngIntElt

      Bounds on the Cardinality of a Largest Code
            EliasBound(K, n, d) : FldFin, RngIntElt, RngIntElt -> RngIntElt
            GriesmerBound(K, n, d) : FldFin, RngIntElt, RngIntElt -> RngIntElt
            JohnsonBound(n, d) : RngIntElt, RngIntElt -> RngIntElt
            LevenshteinBound(K, n, d) : FldFin, RngIntElt, RngIntElt -> RngIntElt
            PlotkinBound(K, n, d) : FldFin, RngIntElt, RngIntElt -> RngIntElt
            SingletonBound(K, n, d) : FldFin, RngIntElt, RngIntElt -> RngIntElt
            SpherePackingBound(K, n, d) : FldFin, RngIntElt, RngIntElt -> RngIntElt
            GilbertVarshamovBound(K, n, d) : FldFin, RngIntElt, RngIntElt -> RngIntElt
            GilbertVarshamovLinearBound(K, n, d) : FldFin, RngIntElt, RngIntElt -> RngIntElt
            VanLintBound(K, n, d) : FldFin, RngIntElt, RngIntElt -> RngIntElt
            Example CodeFld_Card-Best-Comparison (H152E37)

      Bounds on the Minimum Distance
            BCHBound(C) : Code -> RngIntElt, RngIntElt
            GriesmerMinimumWeightBound(K, n, k) : FldFin, RngIntElt, RngIntElt->RngIntElt

      Asymptotic Bounds on the Information Rate
            EliasAsymptoticBound(K, delta) : FldFin, FldPrElt -> FldPrElt
            McElieceEtAlAsymptoticBound(delta) : FldPrElt -> FldPrElt
            PlotkinAsymptoticBound(K, delta) : FldFin, FldPrElt -> FldPrElt
            SingletonAsymptoticBound(delta) : FldPrElt -> FldPrElt
            HammingAsymptoticBound(K, delta) : FldFin, FldPrElt -> FldPrElt
            GilbertVarshamovAsymptoticBound(K, delta) : FldFin, FldPrElt -> FldPrElt

      Other Bounds
            GriesmerLengthBound(K, k, d) : FldFin, RngIntElt, RngIntElt -> RngIntElt

 
Best Known Linear Codes
      BKLC(K, n, k) : FldFin, RngIntElt, RngIntElt -> Code, BoolElt
      BLLC(K, k, d) : FldFin, RngIntElt, RngIntElt -> Code, BoolElt
      BDLC(K, n, d) : FldFin, RngIntElt, RngIntElt -> Code, BoolElt
      Example CodeFld_BKLC-GF2 (H152E38)
      Example CodeFld_BKLC-GF4 (H152E39)
      Example CodeFld_BestLength-GF2 (H152E40)
      Example CodeFld_BDLC-GF4 (H152E41)
      Example CodeFld_VerboseBestCode (H152E42)

 
Decoding
      Decode(C, v: parameters) : Code, ModTupFldElt -> BoolElt, ModTupFldElt
      Decode(C, Q: parameters) : Code, [ ModTupFldElt ] -> [ BoolElt ], [ ModTupFldElt ]
      Example CodeFld_Decode (H152E43)

 
Transforms

      Mattson--Solomon Transforms
            MattsonSolomonTransform(f, n) : RngUPolElt, RngIntElt -> RngUPolElt
            InverseMattsonSolomonTransform(A, n) : RngUPolElt, RngIntElt -> RngUPolElt
            Example CodeFld_MattsonSolomonTransform (H152E44)

      Krawchouk Polynomials
            KrawchoukPolynomial(K, n, k) : FldFin, RngIntElt, RngIntElt -> RngUPolElt
            KrawchoukTransform(f, K, n) : RngUPolElt, FldFin, RngIntElt -> RngUPolElt
            InverseKrawchouk(A, K, n) : RngUPolElt, FldFin, RngIntElt -> RngUPolElt

 
Automorphism Groups

      Introduction

      Group Actions
            v ^ x : ModTupFldElt, GrpPermElt -> ModTupFldElt
            v ^ G : ModTupFldElt, GrpPerm -> GSet{ ModTupFldElt }
            C ^ x : Code, GrpPermElt -> Code
            C ^ G : Code, GrpPerm -> GSet{ Code }
            S ^ x : { ModTupFldElt }, GrpPermElt -> { ModTupFldElt }
            S ^ x : { Code }, GrpPermElt -> { Code }
            Fix(C, G) : Code, GrpPerm -> Code

      Automorphism Group
            AutomorphismGroup(C: parameters) : Code -> GrpPerm, PowMap, Map
            PermutationGroup(C) : Code -> GrpPerm, PowMap, Map
            AutomorphismSubgroup(C) : Code -> GrpPerm, PowMap, Map
            AutomorphismGroupStabilizer(C, k) : Code, RngIntElt -> GrpPerm, PowMap, Map
            Aut(C) : Code -> Pow, Map
            Aut(C, T) : Code, MonStgElt -> Pow, Map
            Example CodeFld_AutomorphismGroup (H152E45)
            Example CodeFld_AutoMorphismGroupWithWeight (H152E46)

      Equivalence and Isomorphism of Codes
            IsIsomorphic(C, D: parameters) : Code, Code -> BoolElt, Map

 
Bibliography

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Version: V2.19 of Mon Dec 17 14:40:36 EST 2012