Additive Codes

• Introduction
• Construction of Additive Codes
  • Construction of General Additive Codes
    • AdditiveCode<F, K, n | L>: FldFin, RngIntElt, List → Code
    • AdditiveCode(G): ModMatRngElt → CodeAdd
    • AdditiveCode(K, G): FldFin, ModMatRngElt → CodeAdd
    • Example: Linear and Additive Codes Compared
    • Example: Additive Code from a Matrix
    • AdditiveCode(K, C): FldFin, Code → CodeAdd
    • Example: Additive Code from a Linear Code
    • Example: Viewing an Additive Code over a Subfield
  • 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: Chain of Trivial Additive Codes
• Invariants of an Additive Code
  • The Ambient Space and Alphabet
    • Alphabet(C): Code → Rng
    • Field(C): Code → Rng
    • CoefficientField(C): Code → Rng
    • AmbientSpace(C): Code → ModTupRng
    • Generic(C): Code → Code
    • Example: Alphabet and Coefficient Field
  • Basic Numerical Invariants
    • Length(C): Code → RngIntElt
    • Dimension(C): Code → FldRatElt
    • NumberOfGenerators(C): Code → RngIntElt
    • Ngens(C): Code → RngIntElt
    • # C: Code → RngIntElt
    • InformationRate(C): Code → FldPrElt
  • The Code Space
    • GeneratorMatrix(C): Code → ModMatFldElt
    • BasisMatrix(C): Code → ModMatRngElt
    • Basis(C): Code → [ ModTupRngElt ]
    • Generators(C): Code → { ModTupFldElt }
    • 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, ..., aₙ]: Code, [ RngElt ] → ModTupRngElt
    • elt< C | a₁, ..., aₙ>: Code, List → 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
    • InnerProduct(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
    • Trace(u): ModTupFldElt → 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: Subcode between Two Codes
  • 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
    • MinimumDistance(C: parameters): CodeAdd → RngIntElt
    • Example: Minimum Weight of an Additive Code
  • 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(K, v): FldFin, ModTupFldElt → CodeAdd
    • AdditiveCyclicCode(Q): [ModTupFldElt] → CodeAdd
    • AdditiveCyclicCode(K, Q): FldFin, [ModTupFldElt] → CodeAdd
    • AdditiveCyclicCode(n, f): RngIntElt, RngUPolElt → CodeAdd
    • AdditiveCyclicCode(K, n, f): FldFin, RngIntElt, RngUPolElt → CodeAdd
    • AdditiveCyclicCode(n, Q): RngIntElt, [RngUPolElt] → CodeAdd
    • AdditiveCyclicCode(K, n, Q): FldFin, 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(K, n, Q): FldFin, RngIntElt, SeqEnum[RngUPolElt] → CodeAdd
    • AdditiveQuasiCyclicCode(n, Q, h): RngIntElt, SeqEnum[RngUPolElt], RngIntElt → CodeAdd
    • AdditiveQuasiCyclicCode(K, n, Q, h): FldFin, RngIntElt, SeqEnum[RngUPolElt], RngIntElt → CodeAdd
    • AdditiveQuasiCyclicCode(Q): SeqEnum[ModTupFldElt] → CodeAdd
    • AdditiveQuasiCyclicCode(K, Q): FldFin, SeqEnum[ModTupFldElt] → CodeAdd
    • AdditiveQuasiCyclicCode(Q, h): SeqEnum[ModTupFldElt], RngIntElt → CodeAdd
    • AdditiveQuasiCyclicCode(K, Q, h): FldFin, 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