# Additive Codes - [Introduction](introduction.md) - [Construction of Additive Codes](creation.md) - [Construction of General Additive Codes](creation.md#construction-of-general-additive-codes) - [`AdditiveCode: FldFin, RngIntElt, List → Code`](creation.md#constructor-constructor-additivecode-fldfin-rngintelt-list-code) - [`AdditiveCode(G): ModMatRngElt → CodeAdd`](creation.md#function-additivecode-modmatrngelt) - [`AdditiveCode(K, G): FldFin, ModMatRngElt → CodeAdd`](creation.md#function-additivecode-fldfin-modmatrngelt) - [`Example: Add Lin Diff`](creation.md#example-ex-f9cba7) - [`Example: Code Add From Matrix`](creation.md#example-ex-9d93b8) - [`AdditiveCode(K, C): FldFin, Code → CodeAdd`](creation.md#function-additivecode-fldfin-code) - [`Example: Code Add From Code`](creation.md#example-ex-8b74a8) - [`Example: Code Add From Code Fail`](creation.md#example-ex-770cac) - [Some Trivial Additive Codes](creation.md#some-trivial-additive-codes) - [`AdditiveZeroCode(F, K, n): FldFin, FldFin, RngIntElt → CodeAdd`](creation.md#function-additivezerocode-fldfin-fldfin-rngintelt) - [`AdditiveRepetitionCode(F, K, n): FldFin, FldFin, RngIntElt → Code`](creation.md#function-additiverepetitioncode-fldfin-fldfin-rngintelt) - [`AdditiveZeroSumCode(F, K, n): FldFin, FldFin, RngIntElt → Code`](creation.md#function-additivezerosumcode-fldfin-fldfin-rngintelt) - [`AdditiveUniverseCode(F, K, n): FldFin, FldFin, RngIntElt → Code`](creation.md#function-additiveuniversecode-fldfin-fldfin-rngintelt) - [`RandomAdditiveCode(F, K, n, k): FldFin, FldFin, RngIntElt, RngIntElt → Code`](creation.md#function-randomadditivecode-fldfin-fldfin-rngintelt-rngintelt) - [`Example: Simple Code Chain`](creation.md#example-ex-24f23c) - [Invariants of an Additive Code](access.md) - [The Ambient Space and Alphabet](access.md#the-ambient-space-and-alphabet) - [`Alphabet(C): Code → Rng`](access.md#function-alphabet-code) - [`Field(C): Code → Rng`](access.md#function-field-code) - [`CoefficientField(C): Code → Rng`](access.md#function-coefficientfield-code) - [`AmbientSpace(C): Code → ModTupRng`](access.md#function-ambientspace-code) - [`Generic(C): Code → Code`](access.md#function-generic-code) - [`Example: Alphabet Coeff Field`](access.md#example-ex-b0a75c) - [Basic Numerical Invariants](access.md#basic-numerical-invariants) - [`Length(C): Code → RngIntElt`](access.md#function-length-code) - [`Dimension(C): Code → FldRatElt`](access.md#function-dimension-code) - [`NumberOfGenerators(C): Code → RngIntElt`](access.md#function-numberofgenerators-code) - [`Ngens(C): Code → RngIntElt`](access.md#function-ngens-code) - [`# C: Code → RngIntElt`](access.md#operation-operation-code-rngintelt) - [`InformationRate(C): Code → FldPrElt`](access.md#function-informationrate-code) - [The Code Space](access.md#the-code-space) - [`GeneratorMatrix(C): Code → ModMatFldElt`](access.md#function-generatormatrix-code) - [`BasisMatrix(C): Code → ModMatRngElt`](access.md#function-basismatrix-code) - [`Basis(C): Code → [ ModTupRngElt ]`](access.md#function-basis-code) - [`Generators(C): Code → { ModTupFldElt }`](access.md#function-generators-code) - [`C . i: Code, RngIntElt → ModTupFldElt`](access.md#operation-operation-code-rngintelt-modtupfldelt) - [The Dual Space](access.md#the-dual-space) - [`Dual(C): CodeAdd → CodeAdd`](access.md#function-dual-codeadd) - [`ParityCheckMatrix(C): Code → ModMatFldElt`](access.md#function-paritycheckmatrix-code) - [Operations on Codewords](element.md) - [Construction of a Codeword](element.md#construction-of-a-codeword) - [`C ! [a₁, ..., aₙ]: Code, [ RngElt ] → ModTupRngElt`](element.md#operation-op-code-rngelt) - [`elt< C | a₁, ..., aₙ>: Code, List → ModTupRngElt`](element.md#constructor-constructor-elt-code-list-modtuprngelt) - [`C ! u: Code, ModTupRngElt → ModTupRngElt`](element.md#operation-op-code-modtuprngelt) - [`C ! 0: Code, RngIntElt → ModTupRngElt`](element.md#operation-op-code-rngintelt) - [`Random(C): Code → ModTupRngElt`](element.md#function-random-code) - [Arithmetic Operations on Codewords](element.md#arithmetic-operations-on-codewords) - [`u + v: ModTupRngElt, ModTupRngElt → ModTupRngElt`](element.md#operation-op-plus-modtuprngelt-modtuprngelt) - [`- u: ModTupRngElt → ModTupRngElt`](element.md#operation-operation-modtuprngelt-modtuprngelt) - [`u - v: ModTupRngElt, ModTupRngElt → ModTupRngElt`](element.md#operation-op-minus-modtuprngelt-modtuprngelt) - [`a * u: RngElt, ModTupRngElt → ModTupRngElt`](element.md#operation-op-times-rngelt-modtuprngelt) - [`Normalize(u): ModTupRngElt → ModTupRngElt`](element.md#function-normalize-modtuprngelt) - [Distance and Weight](element.md#distance-and-weight) - [`Distance(u, v): ModTupRngElt, ModTupRngElt → RngIntElt`](element.md#function-distance-modtuprngelt-modtuprngelt) - [`Weight(u): ModTupRngElt → RngIntElt`](element.md#function-weight-modtuprngelt) - [Vector Space and Related Operations](element.md#vector-space-and-related-operations) - [`(u, v): ModTupRngElt, ModTupRngElt → RngElt`](element.md#literal-literal-u-v-modtuprngelt-modtuprngelt-rngelt) - [`InnerProduct(u, v): ModTupRngElt, ModTupRngElt → RngElt`](element.md#function-innerproduct-modtuprngelt-modtuprngelt) - [`TraceInnerProduct(K, u, v): FldFin, ModTupFldElt, ModTupFldElt → FldFinElt`](element.md#function-traceinnerproduct-fldfin-modtupfldelt-modtupfldelt) - [`Support(w): ModTupRngElt → { RngIntElt }`](element.md#function-support-modtuprngelt) - [`Coordinates(C, u): Code, ModTupRngElt → [ RngFinElt ]`](element.md#function-coordinates-code-modtuprngelt) - [`Parent(w): ModTupRngElt → ModTupRng`](element.md#function-parent-modtuprngelt) - [`Rotate(u, k): ModTupRngElt, RngIntElt → ModTupRngElt`](element.md#function-rotate-modtuprngelt-rngintelt) - [`Rotate(~u, k): ModTupRngElt, RngIntElt`](element.md#function-rotate-modtuprngelt-rngintelt-ref) - [`Trace(u, S): ModTupFldElt, FldFin → ModTupFldElt`](element.md#function-trace-modtupfldelt-fldfin) - [`Trace(u): ModTupFldElt → ModTupFldElt`](element.md#function-trace-modtupfldelt) - [Predicates for Codewords](element.md#predicates-for-codewords) - [`u eq v: ModTupRngElt, ModTupRngElt → BoolElt`](element.md#operation-op-eq-modtuprngelt-modtuprngelt) - [`u ne v: ModTupRngElt, ModTupRngElt → BoolElt`](element.md#operation-op-ne-modtuprngelt-modtuprngelt) - [`IsZero(u): ModTupRngElt → BoolElt`](element.md#function-iszero-modtuprngelt) - [Accessing Components of a Codeword](element.md#accessing-components-of-a-codeword) - [`u[i]: ModTupRngElt, RngIntElt → RngElt`](element.md#literal-literal-u-i-modtuprngelt-rngintelt-rngelt) - [`u[i] := x;`](element.md#literal-literal-u-i-x) - [Subcodes](subcode.md) - [The Subcode Constructor](subcode.md#the-subcode-constructor) - [`sub: CodeAdd, List → CodeAdd`](subcode.md#constructor-constructor-sub-codeadd-list-codeadd) - [`Subcode(C, k): CodeAdd, RngIntElt → CodeAdd`](subcode.md#function-subcode-codeadd-rngintelt) - [`Subcode(C, S): CodeAdd, { RngIntElt } → Code`](subcode.md#function-subcode-codeadd-rngintelt-2) - [`SubcodeBetweenCode(C1, C2, k): CodeAdd, CodeAdd, RngIntElt → CodeAdd`](subcode.md#function-subcodebetweencode-codeadd-codeadd-rngintelt) - [`SubcodeWordsOfWeight(C, w): CodeAdd, RngIntElt → CodeAdd`](subcode.md#function-subcodewordsofweight-codeadd-rngintelt) - [`SubcodeWordsOfWeight(C, S): CodeAdd, { RngIntElt } → CodeAdd`](subcode.md#function-subcodewordsofweight-codeadd-rngintelt-2) - [`Example: Subcode Between Code`](subcode.md#example-ex-18722c) - [Sum, Intersection and Dual](subcode.md#sum-intersection-and-dual) - [`C + D: CodeAdd, CodeAdd → CodeAdd`](subcode.md#operation-op-plus-codeadd-codeadd) - [`C meet D: Code, Code → Code`](subcode.md#operation-op-meet-code-code) - [`Dual(C): Code → Code`](subcode.md#function-dual-code) - [Membership and Equality](subcode.md#membership-and-equality) - [`u in C: ModTupRngElt, CodeAdd → BoolElt`](subcode.md#operation-op-in-modtuprngelt-codeadd) - [`u notin C: ModTupRngElt, CodeAdd → BoolElt`](subcode.md#operation-op-notin-modtuprngelt-codeadd) - [`C subset D: Code, Code → BoolElt`](subcode.md#operation-op-subset-code-code) - [`C notsubset D: Code, Code → BoolElt`](subcode.md#operation-operation-notsubset-code-code-boolelt) - [`C eq D: Code, Code → BoolElt`](subcode.md#operation-op-eq-code-code) - [`C ne D: Code, Code → BoolElt`](subcode.md#operation-op-ne-code-code) - [Properties of Codes](Boolean.md) - [`IsSelfDual(C): Code → BoolElt`](Boolean.md#function-isselfdual-code) - [`IsSelfOrthogonal(C): Code → BoolElt`](Boolean.md#function-isselforthogonal-code) - [`IsPerfect(C): Code → BoolElt`](Boolean.md#function-isperfect-code) - [`IsProjective(C): Code → BoolElt`](Boolean.md#function-isprojective-code) - [`IsAdditiveProjective(C): CodeAdd → BoolElt`](Boolean.md#function-isadditiveprojective-codeadd) - [The Weight Distribution](structural.md) - [The Minimum Weight](structural.md#the-minimum-weight) - [`MinimumWeight(C: parameters): CodeAdd → RngIntElt`](structural.md#function-minimumweight-codeadd) - [`MinimumDistance(C: parameters): CodeAdd → RngIntElt`](structural.md#function-minimumdistance-codeadd) - [`Example: Additive Minweight`](structural.md#example-ex-1dafbe) - [The Weight Distribution](structural.md#id1) - [`WeightDistribution(C): CodeAdd → [ ]`](structural.md#function-weightdistribution-codeadd) - [`DualWeightDistribution(C): CodeAdd → [ ]`](structural.md#function-dualweightdistribution-codeadd) - [The Weight Enumerator](structural.md#the-weight-enumerator) - [`WeightEnumerator(C): CodeAdd → RngMPolElt`](structural.md#function-weightenumerator-codeadd) - [`CompleteWeightEnumerator(C): CodeAdd → RngMPolElt`](structural.md#function-completeweightenumerator-codeadd) - [`CompleteWeightEnumerator(C, u): Code, ModTupFldElt → RngMPolElt`](structural.md#function-completeweightenumerator-code-modtupfldelt) - [The MacWilliams Transform](structural.md#the-macwilliams-transform) - [`MacWilliamsTransform(n, k, q, W): RngIntElt, RngIntElt, RngIntElt, [ ] → [ ]`](structural.md#function-macwilliamstransform-rngintelt-rngintelt-rngintelt-rngintelt-rngintelt) - [Words](structural.md#words) - [`Words(C, w: parameters): Code, RngIntElt → { ModTupFldElt }`](structural.md#function-words-code-rngintelt) - [`NumberOfWords(C, w): Code, RngIntElt → RngIntElt`](structural.md#function-numberofwords-code-rngintelt) - [`WordsOfBoundedWeight(C, l, u: parameters): Code, RngIntElt, RngIntElt → { ModTupFldElt }`](structural.md#function-wordsofboundedweight-code-rngintelt-rngintelt) - [Families of Linear Codes](families.md) - [Cyclic Codes](families.md#cyclic-codes) - [`AdditiveCyclicCode(v): ModTupFldElt → CodeAdd`](families.md#function-additivecycliccode-modtupfldelt) - [`AdditiveCyclicCode(K, v): FldFin, ModTupFldElt → CodeAdd`](families.md#function-additivecycliccode-fldfin-modtupfldelt) - [`AdditiveCyclicCode(Q): [ModTupFldElt] → CodeAdd`](families.md#function-additivecycliccode-modtupfldelt-2) - [`AdditiveCyclicCode(K, Q): FldFin, [ModTupFldElt] → CodeAdd`](families.md#function-additivecycliccode-fldfin-modtupfldelt-2) - [`AdditiveCyclicCode(n, f): RngIntElt, RngUPolElt → CodeAdd`](families.md#function-additivecycliccode-rngintelt-rngupolelt) - [`AdditiveCyclicCode(K, n, f): FldFin, RngIntElt, RngUPolElt → CodeAdd`](families.md#function-additivecycliccode-fldfin-rngintelt-rngupolelt) - [`AdditiveCyclicCode(n, Q): RngIntElt, [RngUPolElt] → CodeAdd`](families.md#function-additivecycliccode-rngintelt-rngupolelt-2) - [`AdditiveCyclicCode(K, n, Q): FldFin, RngIntElt, [RngUPolElt] → CodeAdd`](families.md#function-additivecycliccode-fldfin-rngintelt-rngupolelt-2) - [`AdditiveCyclicCode(v4, v2): ModTupFldElt, ModTupFldElt → CodeAdd`](families.md#function-additivecycliccode-modtupfldelt-modtupfldelt) - [`AdditiveCyclicCode(n, f4, f2): RngIntElt, RngUPolElt, RngUPolElt → CodeAdd`](families.md#function-additivecycliccode-rngintelt-rngupolelt-rngupolelt) - [Quasicyclic Codes](families.md#quasicyclic-codes) - [`AdditiveQuasiCyclicCode(n, Q): RngIntElt, SeqEnum[RngUPolElt] → CodeAdd`](families.md#function-additivequasicycliccode-rngintelt-seqenum-rngupolelt) - [`AdditiveQuasiCyclicCode(K, n, Q): FldFin, RngIntElt, SeqEnum[RngUPolElt] → CodeAdd`](families.md#function-additivequasicycliccode-fldfin-rngintelt-seqenum-rngupolelt) - [`AdditiveQuasiCyclicCode(n, Q, h): RngIntElt, SeqEnum[RngUPolElt], RngIntElt → CodeAdd`](families.md#function-additivequasicycliccode-rngintelt-seqenum-rngupolelt-rngintelt) - [`AdditiveQuasiCyclicCode(K, n, Q, h): FldFin, RngIntElt, SeqEnum[RngUPolElt], RngIntElt → CodeAdd`](families.md#function-additivequasicycliccode-fldfin-rngintelt-seqenum-rngupolelt-rngintelt) - [`AdditiveQuasiCyclicCode(Q): SeqEnum[ModTupFldElt] → CodeAdd`](families.md#function-additivequasicycliccode-seqenum-modtupfldelt) - [`AdditiveQuasiCyclicCode(K, Q): FldFin, SeqEnum[ModTupFldElt] → CodeAdd`](families.md#function-additivequasicycliccode-fldfin-seqenum-modtupfldelt) - [`AdditiveQuasiCyclicCode(Q, h): SeqEnum[ModTupFldElt], RngIntElt → CodeAdd`](families.md#function-additivequasicycliccode-seqenum-modtupfldelt-rngintelt) - [`AdditiveQuasiCyclicCode(K, Q, h): FldFin, SeqEnum[ModTupFldElt], RngIntElt → CodeAdd`](families.md#function-additivequasicycliccode-fldfin-seqenum-modtupfldelt-rngintelt) - [New Codes from Old](operation.md) - [Standard Constructions](operation.md#standard-constructions) - [`AugmentCode(C): CodeAdd → Code`](operation.md#function-augmentcode-codeadd) - [`CodeComplement(C, S): Code, Code → Code`](operation.md#function-codecomplement-code-code) - [`DirectSum(C, D): Code, Code → Code`](operation.md#function-directsum-code-code) - [`DirectSum(Q): [Code] → Code`](operation.md#function-directsum-code) - [`DirectProduct(C, D): Code, Code → Code`](operation.md#function-directproduct-code-code) - [`ExtendCode(C): CodeAdd → CodeAdd`](operation.md#function-extendcode-codeadd) - [`ExtendCode(C, n): Code, RngIntElt → Code`](operation.md#function-extendcode-code-rngintelt) - [`PadCode(C, n): CodeAdd, RngIntElt → CodeAdd`](operation.md#function-padcode-codeadd-rngintelt) - [`PlotkinSum(C1, C2): Code, Code → Code`](operation.md#function-plotkinsum-code-code) - [`PlotkinSum(C1, C2, C3: parameters): Code, Code, Code → Code`](operation.md#function-plotkinsum-code-code-code) - [`PunctureCode(C, i): CodeAdd, RngIntElt → Code`](operation.md#function-puncturecode-codeadd-rngintelt) - [`PunctureCode(C, S): CodeAdd, { RngIntElt } → Code`](operation.md#function-puncturecode-codeadd-rngintelt-2) - [`ShortenCode(C, i): Code, RngIntElt → Code`](operation.md#function-shortencode-code-rngintelt) - [`ShortenCode(C, S): Code, { RngIntElt } → Code`](operation.md#function-shortencode-code-rngintelt-2) - [Combining Codes](operation.md#combining-codes) - [`C1 cat C2: CodeAdd, CodeAdd → CodeAdd`](operation.md#operation-operation-cat-codeadd-codeadd-codeadd) - [`Juxtaposition(C1, C2): Code, Code → Code`](operation.md#function-juxtaposition-code-code) - [Automorphism Group](additive-automorphisms.md) - [`AutomorphismGroup(C): CodeAdd → GrpPerm`](additive-automorphisms.md#function-automorphismgroup-codeadd) - [`PermutationGroup(C): CodeAdd → GrpPerm`](additive-automorphisms.md#function-permutationgroup-codeadd)