Linear Codes over Finite Rings

• Introduction
• Constructions
  • 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: Ternary Golay Code
    • Example: Code From Matrix
    • Example: Permutation Code
  • 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
    • RandomLinearCode(R, n, k): Rng, RngIntElt, RngIntElt → Code
    • Example: Simple Finite Ring
  • General Cyclic Codes
    • CyclicCode(u): ModTupRngElt → Code
    • CyclicCode(n, g): RngIntElt, RngUPolElt → Code
    • CyclotomicFactors(R, n): Rng, RngIntElt → [RngUPolElt]
    • Example: Cyclic Code
    • Example: Cyclic Galois Ring
• Invariants
  • # C: Code → RngIntElt
  • C . i: Code, RngIntElt → ModTupRngElt
  • Name(C, i): Code, RngIntElt → ModTupRngElt
  • Alphabet(C): Code → Rng
  • AmbientSpace(C): Code → ModTupRng
  • Basis(C): Code → [ ModTupRngElt ]
  • Generators(C): Code → { ModTupRngElt }
  • GeneratorMatrix(C): Code → ModMatRngElt
  • Generic(C): Code → Code
  • Length(C): Code → RngIntElt
  • PseudoDimension(C): Code → RngIntElt
  • NumberOfGenerators(C): Code → RngIntElt
  • Ngens(C): Code → RngIntElt
  • ParityCheckMatrix(C): Code → ModMatRngElt
  • Random(C): Code → ModTupRngElt
  • RSpace(C): Code → ModTupRng
  • InformationRate(C): Code → RngPrElt
• Subcodes
  • The Subcode Constructor
    • sub<C | L>: Code, List → Code
    • Subcode(C, t): Code, RngIntElt → Code
    • Subcode(C, S): Code, {RngIntElt} → Code
    • Example: Subcode Galois Rings
• 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
  • IsSelfOrthogonal(C): Code → BoolElt
  • IsProjective(C): Code → BoolElt
  • IsZero(u): ModTupRngElt → BoolElt
  • Example: Self Dual Z4
• New Codes from Old
  • Sum, Intersection and Dual
    • C + D: Code, Code → Code
    • C meet D: Code, Code → Code
    • Dual(C): Code → Code
    • Example: Sum Intersection
  • Standard Constructions
    • DirectSum(C, D): Code, Code → Code
    • DirectProduct(C, D): Code, Code → Code
    • C1 cat C2: 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
    • Example: lengths
    • Example: Punct Z4
• Codeword Operations
  • Construction
    • C ! [a₁, ..., aₙ]: Code, [ RngElt ] → ModTupRngElt
    • elt< C | a₁, ..., aₙ>: Code, List → ModTupRngElt
    • C ! u: Code, ModTupRngElt → ModTupRngElt
    • C ! 0: Code, RngIntElt → ModTupRngElt
    • Example: Code Elts
  • Operations
    • 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
    • InnerProduct(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
    • Example: Codeword Ops
  • Accessing Components of a Codeword
    • u[i]: ModTupRngElt, RngIntElt → RngElt
    • u[i] := x;
• Weight Distributions
  • Hamming Weight
    • MinimumWeight(C): Code → RngIntElt
    • MinimumDistance(C): Code → RngIntElt
    • WeightDistribution(C): Code → [ <RngIntElt, RngIntElt> ]
    • DualWeightDistribution(C): Code → [ <RngIntElt, RngIntElt> ]
    • Example: Weight Dist Cyc
• Weight Enumerators
  • CompleteWeightEnumerator(C): Code → RngMPolElt
  • WeightEnumerator(C): Code → RngMPolElt
  • HammingWeightEnumerator(C): Code → RngMPolElt
  • Example: WeightEnum Galois Rings