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Acknowledgements
Introduction
Construction of Codes
Construction of General Linear Codes
Construction of Simple Linear Codes
Construction of General Cyclic Codes
Invariants of Codes
Codes over Z4
The Gray Map
Families of Codes over Z4
Derived Binary Codes
The Standard Form
Other Z4 functions
Construction of Subcodes of Linear Codes
The Subcode Constructor
Weight Distributions
Hamming Weight
Lee Weight
Euclidean Weight
Weight Enumerators
Constructing New Codes from Old
Sum, Intersection and Dual
Standard Constructions
Operations on Codewords
Construction of a Codeword
Operations on Codewords and Vectors
Accessing Components of a Codeword
Boolean Predicates
Bibliography
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 (H130E1)
Example CodeRng_CodeFromMatrix (H130E2)
Example CodeRng_PermutationCode (H130E3)
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 (H130E4)
Construction of General Cyclic Codes
CyclicCode(u) : ModTupRngElt -> Code
CyclicCode(n, g) : RngIntElt, RngUPolElt -> Code
CyclotomicFactors(R, n) : Rng, RngIntElt -> [RngUPolElt]
Example CodeRng_CyclicCode (H130E5)
Example CodeRng_cyclic-galois-ring (H130E6)
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
Codes over Z4
The Gray Map
GrayMap(C) : Code -> Map
GrayMapImage(C) : Code -> [ ModTupRngElt ]
HasLinearGrayMapImage(C) : Code -> BoolElt, Code
Example CodeRng_GrayMap (H130E7)
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 (H130E8)
Derived Binary Codes
BinaryResidueCode(C) : Code -> Code
BinaryTorsionCode(C) : Code -> Code
Z4CodeFromBinaryChain(C1, C2) : Code, Code -> Code
Example CodeRng_derived-binary (H130E9)
The Standard Form
StandardForm(C) : Code -> Code, Map
Example CodeRng_StandardForm (H130E10)
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 (H130E11)
Weight Distributions
Hamming Weight
MinimumWeight(C) : Code -> RngIntElt
WeightDistribution(C) : Code -> [ <RngIntElt, RngIntElt> ]
DualWeightDistribution(C) : Code -> [ <RngIntElt, RngIntElt> ]
Example CodeRng_weight-dist-cyc (H130E12)
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 (H130E13)
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 (H130E14)
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 (H130E15)
Example CodeRng_WeightEnumerator (H130E16)
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 (H130E17)
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 (H130E18)
Example CodeRng_punct-z4 (H130E19)
Operations on Codewords
Construction of a Codeword
C ! [a1, ..., an] : Code, [ RngElt ] -> ModTupRngElt
C ! u : Code, ModTupRngElt -> ModTupRngElt
C ! 0 : Code, RngIntElt -> ModTupRngElt
Example CodeRng_code-elts (H130E20)
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 (H130E21)
Accessing Components of a Codeword
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
IsZero(u) : ModTupRngElt -> BoolElt
Bibliography
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