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 (H155E1)
Example CodeRng_CodeFromMatrix (H155E2)
Example CodeRng_PermutationCode (H155E3)
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
RandomLinearCode(R, n, k) : Rng, RngIntElt, RngIntElt -> Code
Example CodeRng_simple-finite-ring (H155E4)
Construction of General Cyclic Codes
CyclicCode(u) : ModTupRngElt -> Code
CyclicCode(n, g) : RngIntElt, RngUPolElt -> Code
CyclotomicFactors(R, n) : Rng, RngIntElt -> [RngUPolElt]
Example CodeRng_CyclicCode (H155E5)
Example CodeRng_cyclic-galois-ring (H155E6)
Invariants of Codes
# 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
The Gray Map
GrayMap(C) : Code -> Map
GrayMapImage(C) : Code -> [ ModTupRngElt ]
HasLinearGrayMapImage(C) : Code -> BoolElt, Code
Example CodeRng_GrayMap (H155E7)
Families of Codes over Z4
KerdockCode(m): RngIntElt, RngUPolElt -> Code
PreparataCode(m): RngIntElt, RngUPolElt -> Code
ReedMullerCodeZ4(r, m) : RngIntElt, RngIntElt -> Code
GoethalsCode(m) : RngIntElt -> Code
DelsarteGoethalsCode(m, delta) : RngIntElt, RngIntElt -> Code
GoethalsDelsarteCode(m, delta) : RngIntElt, RngIntElt -> Code
QRCodeZ4(p) : RngIntElt -> Code
GolayCodeZ4(e) : BoolElt -> Code
SimplexAlphaCodeZ4(k) : RngIntElt -> Code
SimplexBetaCodeZ4(k) : RngIntElt -> Code
Example CodeRng_Kerdock (H155E8)
HadamardCodeZ4(δ, m) : RngIntElt, RngIntElt -> CodeLinRng, Mtrx
ExtendedPerfectCodeZ4(δ, m) : RngIntElt, RngIntElt -> CodeLinRng, Mtrx
Example CodeRng_spain-Z4-1 (H155E9)
ReedMullerCodeZ4(r, m) : RngIntElt, RngIntElt -> CodeLinRng
ReedMullerCodesLRMZ4(r, m) : RngIntElt, RngIntElt -> SeqEnum
ReedMullerCodeRMZ4(s, r, m) : RngIntElt, RngIntElt, RngIntElt -> CodeLinRng, Mtrx
Example CodeRng_spain-Z4-2 (H155E10)
ReedMullerCodesRMZ4(s, m) : RngIntElt, RngIntElt -> Tup
Example CodeRng_spain-Z4-3 (H155E11)
Derived Binary Codes
BinaryResidueCode(C) : Code -> Code
BinaryTorsionCode(C) : Code -> Code
Z4CodeFromBinaryChain(C1, C2) : Code, Code -> Code
Example CodeRng_derived-binary (H155E12)
The Standard Form
StandardForm(C) : Code -> Code, Map
Example CodeRng_StandardForm (H155E13)
Constructing New Codes from Old
PlotkinSum(A, B) : Mtrx, Mtrx -> Mtrx
PlotkinSum(C, D) : Code, Code -> Code
QuaternaryPlotkinSum(A, B) : Mtrx, Mtrx -> Mtrx
QuaternaryPlotkinSum(C, D) : Code, Code -> Code
BQPlotkinSum(A, B, C) : Mtrx, Mtrx, Mtrx -> Mtrx
BQPlotkinSum(D, E, F) : Code, Code, Code -> Code
DoublePlotkinSum(A, B, C, D) : Mtrx, Mtrx, Mtrx, Mtrx -> Mtrx
DoublePlotkinSum(E, F, G, H) : Code, Code, Code, Code -> Code
DualKroneckerZ4(C) : CodeLinRng -> CodeLinRng
Example CodeRng_spain-Z4-4 (H155E14)
Example CodeRng_spain-Z4-4a (H155E15)
Invariants of Codes over Z4
SpanZ2CodeZ4(C) : CodeLinRng -> CodeLinFld
KernelZ2CodeZ4(C) : CodeLinRng -> CodeLinRng
DimensionOfSpanZ2(C) : CodeLinRng -> RngIntElt
DimensionOfKernelZ2(C) : CodeLinRng -> RngIntElt
Example CodeRng_spain-Z4-5 (H155E16)
Other Z4 functions
Correlation(v) : ModTupRngElt -> RngQuadElt
Construction of Subcodes of Linear Codes
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 (H155E17)
Hamming Weight
MinimumWeight(C) : Code -> RngIntElt
WeightDistribution(C) : Code -> [ <RngIntElt, RngIntElt> ]
DualWeightDistribution(C) : Code -> [ <RngIntElt, RngIntElt> ]
Example CodeRng_weight-dist-cyc (H155E18)
Lee Weight
LeeWeight(a) : RngIntRes -> RngIntElt
LeeWeight(v) : ModTupRngElt -> RngIntElt
LeeDistance(u, v) : ModTupRngElt, ModTupRngElt -> RngIntElt
MinimumLeeWeight(C) : Code -> RngIntElt
LeeWeightDistribution(C) : Code -> SeqEnum
DualLeeWeightDistribution(C) : Code -> SeqEnum
WordsOfLeeWeight(C, w) : Code, RngIntElt -> SetEnum
WordsOfBoundedLeeWeight(C, l, u) : Code, RngIntElt, RngIntElt -> SetEnum
Example CodeRng_lee-dist (H155E19)
Euclidean Weight
EuclideanWeight(a) : RngIntRes -> RngIntElt
EuclideanWeight(v) : ModTupRngElt -> RngIntElt
EuclideanDistance(u, v) : ModTupRngElt, ModTupRngElt -> RngIntElt
MinimumEuclideanWeight(C) : Code -> RngIntElt
EuclideanWeightDistribution(C) : Code -> SeqEnum
DualEuclideanWeightDistribution(C) : Code -> SeqEnum
Example CodeRng_euclidean-dist (H155E20)
Weight Enumerators
CompleteWeightEnumerator(C): Code -> RngMPolElt
SymmetricWeightEnumerator(C): Code -> RngMPolElt
WeightEnumerator(C): Code -> RngMPolElt
LeeWeightEnumerator(C): Code -> RngMPolElt
EuclideanWeightEnumerator(C): Code -> RngMPolElt
Example CodeRng_weightEnum-galois-rings (H155E21)
Example CodeRng_WeightEnumerator (H155E22)
Constructing 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 (H155E23)
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 (H155E24)
Example CodeRng_punct-z4 (H155E25)
Construction of a Codeword
C ! [a1, ..., an] : Code, [ RngElt ] -> ModTupRngElt
C ! u : Code, ModTupRngElt -> ModTupRngElt
C ! 0 : Code, RngIntElt -> ModTupRngElt
Example CodeRng_code-elts (H155E26)
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
Example CodeRng_codeword-ops (H155E27)
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
IsSelfOrthogonal(C) : Code -> BoolElt
IsProjective(C) : Code -> BoolElt
IsZero(u) : ModTupRngElt -> BoolElt
Example CodeRng_SelfDualZ4 (H155E28)
Bibliography
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Version: V2.19 of
Mon Dec 17 14:40:36 EST 2012