LINEAR CODES OVER FINITE RINGS
Acknowledgements
Introduction
Constructions
General Linear Codes
Simple Linear Codes
General Cyclic Codes
Invariants
Subcodes
The Subcode Constructor
Boolean Predicates
New Codes from Old
Sum, Intersection and Dual
Standard Constructions
Codeword Operations
Construction
Operations
Accessing Components of a Codeword
Weight Distributions
Hamming Weight
Weight Enumerators
Bibliography
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 CodeRng_TernaryGolayCode (H168E1)
Example CodeRng_CodeFromMatrix (H168E2)
Example CodeRng_PermutationCode (H168E3)
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 CodeRng_simple-finite-ring (H168E4)
General Cyclic Codes
CyclicCode(u) : ModTupRngElt -> Code
CyclicCode(n, g) : RngIntElt, RngUPolElt -> Code
CyclotomicFactors(R, n) : Rng, RngIntElt -> [RngUPolElt]
Example CodeRng_CyclicCode (H168E5)
Example CodeRng_cyclic-galois-ring (H168E6)
Invariants
# C : Code -> RngIntElt
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
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 CodeRng_subcode-galois-rings (H168E7)
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 CodeRng_SelfDualZ4 (H168E8)
New Codes from Old
Sum, Intersection and Dual
C + D : Code, Code -> Code
C meet D : Code, Code -> Code
Dual(C) : Code -> Code
Example CodeRng_SumIntersection (H168E9)
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 CodeRng_lengths (H168E10)
Example CodeRng_punct-z4 (H168E11)
Codeword Operations
Construction
C ! [a1, ..., an] : Code, [ RngElt ] -> ModTupRngElt
C ! u : Code, ModTupRngElt -> ModTupRngElt
C ! 0 : Code, RngIntElt -> ModTupRngElt
Example CodeRng_code-elts (H168E12)
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
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 CodeRng_codeword-ops (H168E13)
Accessing Components of a Codeword
u[i] : ModTupRngElt, RngIntElt -> RngElt
u[i] := x;
Weight Distributions
Hamming Weight
MinimumWeight(C) : Code -> RngIntElt
WeightDistribution(C) : Code -> [ <RngIntElt, RngIntElt> ]
DualWeightDistribution(C) : Code -> [ <RngIntElt, RngIntElt> ]
Example CodeRng_weight-dist-cyc (H168E14)
Weight Enumerators
CompleteWeightEnumerator(C): Code -> RngMPolElt
WeightEnumerator(C): Code -> RngMPolElt
Example CodeRng_weightEnum-galois-rings (H168E15)
Bibliography
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