LINEAR CODES OVER FINITE RINGS

DETAILS

 
Introduction

 
Construction of Codes

      Construction of General Linear Codes
            LinearCode<R, n | L> : Rng, RngIntElt, List -> Code
            LinearCode(U) : ModTupRng -> Code
            LinearCode(A) : ModMatRngElt -> Code
            PermutationCode(u, G) : ModTupRngElt, GrpPerm -> Code
            Example CodeRng_TernaryGolayCode (H98E1)
            Example CodeRng_CodeFromMatrix (H98E2)
            Example CodeRng_PermutationCode (H98E3)

      Construction of Simple Linear Codes
            ZeroCode(R, n) : Rng, RngIntElt -> Code
            RepetitionCode(R, n) : Rng, RngIntElt -> Code
            ZeroSumCode(R, n) : Rng, RngIntElt -> Code
            UniverseCode(R, n) : Rng, RngIntElt -> Code
            UniverseCode(R, n) : Rng, RngIntElt -> Code

      Construction of General Cyclic Codes
            CyclicCode(u) : ModTupRngElt -> Code
            CyclicCode(n, g) : RngIntElt, RngUPolElt -> Code
            Z4CyclotomicFactors(n) : RngIntElt -> [RngUPolElt]
            Example CodeRng_CyclicCode (H98E4)

 
Invariants of Codes
      # C : Code -> RngIntElt
      C . i : Code, RngIntElt -> ModTupRngElt
      Alphabet(C) : Code -> Rng
      AmbientSpace(C) : Code -> ModTupRng
      Basis(C) : Code -> [ ModTupRngElt ]
      Dimension(C) : Code -> RngIntElt
      Generators(C) : Code -> { ModTupRngElt }
      GeneratorMatrix(C) : Code -> ModMatRngElt
      Generic(C) : Code -> Code
      Length(C) : Code -> RngIntElt
      NumberOfGenerators(C) : Code -> RngIntElt
      ParityCheckMatrix(C) : Code -> ModMatRngElt
      Random(C): Code -> ModTupRngElt
      RSpace(C) : Code -> ModTupRng
      InformationRate(C) : Code -> RngPrElt

 
Construction of Subcodes of Linear Codes

      The Subcode Constructor
            sub<C | L> : Code, List -> Code
            Subcode(C, k) : Code,RngIntElt -> Code
            Subcode(C, S) : Code, RngIntElt -> Code

      Sum, Intersection and Dual
            C + D : Code, Code -> Code
            C meet D : Code, Code -> Code
            Dual(C) : Code -> Code
            Example CodeRng_SumIntersection (H98E5)

 
Constructing New Codes from Old

      Standard Constructions
            DirectSum(C, D) : Code, Code -> Code
            DirectProduct(C, D) : Code, Code -> Code
            ExtendCode(C) : Code -> Code
            ExtendCode(C, n) : Code, RngIntElt -> Code
            PadCode(C, n) : Code, RngIntElt -> Code
            PlotkinSum(C, D) : 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

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

 
Operations on Codewords

      Construction of a Codeword
            C ! [a_1, ..., a_n] : Code, [ RngElt ] -> ModTupRngElt
            C ! u : Code, ModTupRngElt -> ModTupRngElt
            C ! 0 : Code, RngIntElt -> ModTupRngElt

      Operations on Codewords and Vectors
            u + v : ModTupRngElt, ModTupRngElt -> ModTupRngElt
            - u : ModTupRngElt -> ModTupRngElt
            u - v : ModTupRngElt, ModTupRngElt -> ModTupRngElt
            a * u : RngElt, ModTupRngElt -> ModTupRngElt
            Weight(v) : ModTupRngElt -> RngIntElt
            Distance(u, v) : ModTupRngElt, ModTupRngElt -> RngIntElt
            Support(w) : ModTupRngElt -> { RngIntElt }
            (u, v) : ModTupRngElt, ModTupRngElt : -> RngElt
            Coordinates(C, u) : Code, ModTupRngElt -> [ RngFinElt ]
            Normalize(u) : ModTupRngElt -> ModTupRngElt
            Rotate(u, k) : ModTupRngElt, RngIntElt -> ModTupRngElt
            Rotate(~u, k) : ModTupRngElt, RngIntElt ->
            Parent(w): ModTupRngElt -> ModTupRng

      Lee Weights
            LeeWeight(v) : ModTupRngElt -> RngIntElt
            LeeDistance(u, v) : ModTupRngElt, ModTupRngElt -> RngIntElt
            Example CodeRng_Distance (H98E6)

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

 
Boolean Predicates
      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
      IsCyclic(C) : Code -> BoolElt
      IsSelfDual(C) : Code -> BoolElt
      IsWeaklySelfDual(C) : Code -> BoolElt
      IsZero(u) : ModTupRngElt -> BoolElt
      Example CodeRng_SelfDualZ4 (H98E7)

 
The Standard Form
      StandardForm(C) : Code -> Code, Map
      Example CodeRng_StandardForm (H98E8)

 
The Gray Map
      GrayMap(C) : Code -> Map
      GrayMapImage(C) : Code -> [ ModTupRngElt ]
      HasLinearGrayMapImage(C) : Code -> BoolElt, Code
      Example CodeRng_GrayMap (H98E9)

 
The Weight Distribution

      The Minimum Weight
            MinimumWeight(C) : Code -> RngIntElt

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

      Weight Enumerators
            CompleteWeightEnumerator(C): Code -> RngMPolElt
            SymmetricWeightEnumerator(C): Code -> RngMPolElt
            LeeWeightEnumerator(C): Code -> RngMPolElt
            WeightEnumerator(C): Code -> RngMPolElt
            Example CodeRng_WeightEnumerator (H98E10)

 
Families of Linear Codes
      KerdockCode(m, h): RngIntElt, RngUPolElt -> Code
      KerdockCode(m): RngIntElt, RngUPolElt -> Code
      PreparataCode(m, h): RngIntElt, RngUPolElt -> Code
      PreparataCode(m): RngIntElt, RngUPolElt -> Code

 
Bibliography