# Linear Codes over the Integer Residue Ring ${\mathbb{Z}}_4$

- [Introduction](introduction.md)

- [Constructions for ${\mathbb{Z}}_4$ Codes](constructions.md)

  - [The Gray Map](constructions.md#the-gray-map)

    - [`GrayMap(C): Code → Map`](constructions.md#function-graymap-code)

    - [`GrayMapImage(C): Code → [ ModTupRngElt ]`](constructions.md#function-graymapimage-code)

    - [`HasLinearGrayMapImage(C): Code → BoolElt, Code`](constructions.md#function-haslineargraymapimage-code)

    - [`Example: Gray Map`](constructions.md#example-ex-660fb9)

  - [Families of Codes over ${\mathbb{Z}}_4$](constructions.md#families-of-codes-over-mathbb-z-4)

    - [`KerdockCode(m): RngIntElt → Code`](constructions.md#function-kerdockcode-rngintelt)

    - [`PreparataCode(m): RngIntElt → Code`](constructions.md#function-preparatacode-rngintelt)

    - [`ReedMullerCodeZ4(r, m): RngIntElt, RngIntElt → Code`](constructions.md#function-reedmullercodez4-rngintelt-rngintelt)

    - [`GoethalsCode(m): RngIntElt → Code`](constructions.md#function-goethalscode-rngintelt)

    - [`DelsarteGoethalsCode(m, delta): RngIntElt, RngIntElt → Code`](constructions.md#function-delsartegoethalscode-rngintelt-rngintelt)

    - [`GoethalsDelsarteCode(m, delta): RngIntElt, RngIntElt → Code`](constructions.md#function-goethalsdelsartecode-rngintelt-rngintelt)

    - [`QRCodeZ4(p): RngIntElt → Code`](constructions.md#function-qrcodez4-rngintelt)

    - [`GolayCodeZ4(e): BoolElt → Code`](constructions.md#function-golaycodez4-boolelt)

    - [`SimplexAlphaCodeZ4(k): RngIntElt → Code`](constructions.md#function-simplexalphacodez4-rngintelt)

    - [`SimplexBetaCodeZ4(k): RngIntElt → Code`](constructions.md#function-simplexbetacodez4-rngintelt)

    - [`Example: Kerdock`](constructions.md#example-ex-79827c)

    - [`HadamardCodeZ4(δ, m): RngIntElt, RngIntElt → CodeLinRng, Mtrx`](constructions.md#function-hadamardcodez4-rngintelt-rngintelt)

    - [`ExtendedPerfectCodeZ4(δ, m): RngIntElt, RngIntElt → CodeLinRng, Mtrx`](constructions.md#function-extendedperfectcodez4-rngintelt-rngintelt)

    - [`Example: Spain Z4 1`](constructions.md#example-ex-6e1c45)

    - [`ReedMullerCodeZ4(r, m): RngIntElt, RngIntElt → CodeLinRng`](constructions.md#function-reedmullercodez4-rngintelt-rngintelt-2)

    - [`ReedMullerCodeQRMZ4(r, m): RngIntElt, RngIntElt → CodeLinRng`](constructions.md#function-reedmullercodeqrmz4-rngintelt-rngintelt)

    - [`ReedMullerCodesLRMZ4(r, m): RngIntElt, RngIntElt → SeqEnum`](constructions.md#function-reedmullercodeslrmz4-rngintelt-rngintelt)

    - [`ReedMullerCodeRMZ4(s, r, m): RngIntElt, RngIntElt, RngIntElt → CodeLinRng, Mtrx`](constructions.md#function-reedmullercodermz4-rngintelt-rngintelt-rngintelt)

    - [`Example: Spain Z4 2`](constructions.md#example-ex-797694)

    - [`ReedMullerCodesRMZ4(s, m): RngIntElt, RngIntElt → Tup`](constructions.md#function-reedmullercodesrmz4-rngintelt-rngintelt)

    - [`Example: Spain Z4 3`](constructions.md#example-ex-c9bc7a)

  - [Derived Binary Codes](constructions.md#derived-binary-codes)

    - [`BinaryResidueCode(C): Code → Code`](constructions.md#function-binaryresiduecode-code)

    - [`BinaryTorsionCode(C): Code → Code`](constructions.md#function-binarytorsioncode-code)

    - [`Z4CodeFromBinaryChain(C1, C2): Code, Code → Code`](constructions.md#function-z4codefrombinarychain-code-code)

    - [`Example: Derived Binary`](constructions.md#example-ex-08428d)

  - [New Codes from Old](constructions.md#new-codes-from-old)

    - [`PlotkinSum(A, B): Mtrx, Mtrx → Mtrx`](constructions.md#function-plotkinsum-mtrx-mtrx)

    - [`PlotkinSum(C, D): Code, Code → Code`](constructions.md#function-plotkinsum-code-code)

    - [`QuaternaryPlotkinSum(A, B): Mtrx, Mtrx → Mtrx`](constructions.md#function-quaternaryplotkinsum-mtrx-mtrx)

    - [`QuaternaryPlotkinSum(C, D): Code, Code → Code`](constructions.md#function-quaternaryplotkinsum-code-code)

    - [`BQPlotkinSum(A, B, C): Mtrx, Mtrx, Mtrx → Mtrx`](constructions.md#function-bqplotkinsum-mtrx-mtrx-mtrx)

    - [`BQPlotkinSum(D, E, F): Code, Code, Code → Code`](constructions.md#function-bqplotkinsum-code-code-code)

    - [`DoublePlotkinSum(A, B, C, D): Mtrx, Mtrx, Mtrx, Mtrx → Mtrx`](constructions.md#function-doubleplotkinsum-mtrx-mtrx-mtrx-mtrx)

    - [`DoublePlotkinSum(E, F, G, H): Code, Code, Code, Code → Code`](constructions.md#function-doubleplotkinsum-code-code-code-code)

    - [`DualKroneckerZ4(C): CodeLinRng → CodeLinRng`](constructions.md#function-dualkroneckerz4-codelinrng)

    - [`Example: Spain Z4 4`](constructions.md#example-ex-b631eb)

    - [`Example: Spain Z4 4a`](constructions.md#example-ex-c3f759)

- [Invariants](z4-invar.md)

  - [The Standard Form](z4-invar.md#the-standard-form)

    - [`StandardForm(C): Code → Code, Map`](z4-invar.md#function-standardform-code)

    - [`Example: Standard Form`](z4-invar.md#example-ex-48fad2)

    - [`StandardFormDual(C): CodeLinRng → CodeLinRng, Map`](z4-invar.md#function-standardformdual-codelinrng)

  - [Structures Associated with the Gray Map](z4-invar.md#structures-associated-with-the-gray-map)

    - [`MinRowsGeneratorMatrix(C): CodeLinRng → ModMatRngElt`](z4-invar.md#function-minrowsgeneratormatrix-codelinrng)

    - [`MinRowsParityCheckMatrix(C): CodeLinRng → ModMatRngElt`](z4-invar.md#function-minrowsparitycheckmatrix-codelinrng)

    - [`DualZ4(C): CodeLinRng → CodeLinRng`](z4-invar.md#function-dualz4-codelinrng)

    - [`Example: New Invar`](z4-invar.md#example-ex-3b2558)

    - [`SpanZ2CodeZ4(C): CodeLinRng → CodeLinFld`](z4-invar.md#function-spanz2codez4-codelinrng)

    - [`KernelZ2CodeZ4(C): CodeLinRng → CodeLinRng`](z4-invar.md#function-kernelz2codez4-codelinrng)

    - [`KernelCosetRepresentatives(C): CodeLinRng → SeqEnum, SeqEnum`](z4-invar.md#function-kernelcosetrepresentatives-codelinrng)

    - [`DimensionOfSpanZ2(C): CodeLinRng → RngIntElt`](z4-invar.md#function-dimensionofspanz2-codelinrng)

    - [`RankZ2(C): CodeLinRng → RngIntElt`](z4-invar.md#function-rankz2-codelinrng)

    - [`DimensionOfKernelZ2(C): CodeLinRng → RngIntElt`](z4-invar.md#function-dimensionofkernelz2-codelinrng)

    - [`Example: Spain Z4 5`](z4-invar.md#example-ex-bb9d15)

  - [Coset Representatives](z4-invar.md#coset-representatives)

    - [`CosetRepresentatives(C): CodeLinRng → SeqEnum`](z4-invar.md#function-cosetrepresentatives-codelinrng)

    - [`CosetRepresentatives(C, S): CodeLinRng, CodeLinRng → SeqEnum, SeqEnum`](z4-invar.md#function-cosetrepresentatives-codelinrng-codelinrng)

    - [`Example: Spain Z4 7`](z4-invar.md#example-ex-551655)

  - [Information Space and Information Sets](z4-invar.md#information-space-and-information-sets)

    - [`InformationSpace(C): CodeLinRng → ModTupRng, ModTupFld, Map, Map`](z4-invar.md#function-informationspace-codelinrng)

    - [`Example: Spain Z4 9`](z4-invar.md#example-ex-5fc04a)

    - [`InformationSet(C): CodeLinRng → ModTupRng, ModTupFld, Map, Map`](z4-invar.md#function-informationset-codelinrng)

    - [`IsInformationSet(C, I): CodeLinRng, [RngIntElt] → BoolElt, BoolElt`](z4-invar.md#function-isinformationset-codelinrng-rngintelt)

    - [`Example: Spain Z4 10`](z4-invar.md#example-ex-1d41b7)

  - [Syndrome Space and Coset Leaders](z4-invar.md#syndrome-space-and-coset-leaders)

    - [`SyndromeSpace(C): CodeLinRng → ModTupRng, ModTupFld`](z4-invar.md#function-syndromespace-codelinrng)

    - [`Syndrome(u, C): ModTupFldElt, CodeLinRng → ModTupRngElt`](z4-invar.md#function-syndrome-modtupfldelt-codelinrng)

    - [`Syndrome(u, C): ModTupRngElt, CodeLinRng → ModTupRngElt`](z4-invar.md#function-syndrome-modtuprngelt-codelinrng)

    - [`CosetLeaders(C): CodeLinRng → SetIndx, Map`](z4-invar.md#function-cosetleaders-codelinrng)

    - [`Example: Spain Z4 11`](z4-invar.md#example-ex-5bce83)

  - [Miscellaneous Functions](z4-invar.md#miscellaneous-functions)

    - [`Correlation(v): ModTupRngElt → RngQuadElt`](z4-invar.md#function-correlation-modtuprngelt)

- [Weight Distributions](weight-distribution.md)

  - [Hamming Weight](weight-distribution.md#hamming-weight)

    - [`MinimumWeight(C): Code → RngIntElt`](weight-distribution.md#function-minimumweight-code)

    - [`MinimumDistance(C): Code → RngIntElt`](weight-distribution.md#function-minimumdistance-code)

    - [`WeightDistribution(C): Code → [ <RngIntElt, RngIntElt> ]`](weight-distribution.md#function-weightdistribution-code)

    - [`DualWeightDistribution(C): Code → [ <RngIntElt, RngIntElt> ]`](weight-distribution.md#function-dualweightdistribution-code)

    - [`Example: Weight Dist K8`](weight-distribution.md#example-ex-601dad)

  - [Lee Weight](weight-distribution.md#lee-weight)

    - [`LeeWeight(a): RngIntRes → RngIntElt`](weight-distribution.md#function-leeweight-rngintres)

    - [`LeeWeight(v): ModTupRngElt → RngIntElt`](weight-distribution.md#function-leeweight-modtuprngelt)

    - [`LeeDistance(u, v): ModTupRngElt, ModTupRngElt → RngIntElt`](weight-distribution.md#function-leedistance-modtuprngelt-modtuprngelt)

    - [`MinimumLeeWeight(C): Code → RngIntElt`](weight-distribution.md#function-minimumleeweight-code)

    - [`MinimumLeeDistance(C): Code → RngIntElt`](weight-distribution.md#function-minimumleedistance-code)

    - [`LeeWeightDistribution(C): Code → SeqEnum`](weight-distribution.md#function-leeweightdistribution-code)

    - [`DualLeeWeightDistribution(C): Code → SeqEnum`](weight-distribution.md#function-dualleeweightdistribution-code)

    - [`WordsOfLeeWeight(C, w): Code, RngIntElt → SetEnum`](weight-distribution.md#function-wordsofleeweight-code-rngintelt)

    - [`WordsOfBoundedLeeWeight(C, l, u): Code, RngIntElt, RngIntElt → SetEnum`](weight-distribution.md#function-wordsofboundedleeweight-code-rngintelt-rngintelt)

    - [`Example: Lee Dist`](weight-distribution.md#example-ex-587036)

  - [Euclidean Weight](weight-distribution.md#euclidean-weight)

    - [`EuclideanWeight(a): RngIntRes → RngIntElt`](weight-distribution.md#function-euclideanweight-rngintres)

    - [`EuclideanWeight(v): ModTupRngElt → RngIntElt`](weight-distribution.md#function-euclideanweight-modtuprngelt)

    - [`EuclideanDistance(u, v): ModTupRngElt, ModTupRngElt → RngIntElt`](weight-distribution.md#function-euclideandistance-modtuprngelt-modtuprngelt)

    - [`MinimumEuclideanWeight(C): Code → RngIntElt`](weight-distribution.md#function-minimumeuclideanweight-code)

    - [`MinimumEuclideanDistance(C): Code → RngIntElt`](weight-distribution.md#function-minimumeuclideandistance-code)

    - [`EuclideanWeightDistribution(C): Code → SeqEnum`](weight-distribution.md#function-euclideanweightdistribution-code)

    - [`DualEuclideanWeightDistribution(C): Code → SeqEnum`](weight-distribution.md#function-dualeuclideanweightdistribution-code)

    - [`Example: Euclidean Dist`](weight-distribution.md#example-ex-9f8213)

- [Weight Enumerators](weight-enumerator.md)

  - [`CompleteWeightEnumerator(C): Code → RngMPolElt`](weight-enumerator.md#function-completeweightenumerator-code)

  - [`SymmetricWeightEnumerator(C): Code → RngMPolElt`](weight-enumerator.md#function-symmetricweightenumerator-code)

  - [`WeightEnumerator(C): Code → RngMPolElt`](weight-enumerator.md#function-weightenumerator-code)

  - [`HammingWeightEnumerator(C): Code → RngMPolElt`](weight-enumerator.md#function-hammingweightenumerator-code)

  - [`LeeWeightEnumerator(C): Code → RngMPolElt`](weight-enumerator.md#function-leeweightenumerator-code)

  - [`EuclideanWeightEnumerator(C): Code → RngMPolElt`](weight-enumerator.md#function-euclideanweightenumerator-code)

  - [`Example: Weight Enumerator`](weight-enumerator.md#example-ex-92ee64)

- [Decoding](decoding.md)

  - [Coset Decoding](decoding.md#coset-decoding)

    - [`CosetDecode(C, u : parameters): CodeLinRng, ModTupFldElt → BoolElt, ModTupRngElt, ModTupFldElt`](decoding.md#function-cosetdecode-codelinrng-modtupfldelt)

    - [`CosetDecode(C, u : parameters): CodeLinRng, ModTupRngElt → BoolElt, ModTupRngElt, ModTupFldElt`](decoding.md#function-cosetdecode-codelinrng-modtuprngelt)

    - [`CosetDecode(C, Q : parameters): CodeLinRng, [ModTupFldElt] → SeqEnum, SeqEnum, SeqEnum`](decoding.md#function-cosetdecode-codelinrng-modtupfldelt-2)

    - [`CosetDecode(C, Q : parameters): CodeLinRng, [ModTupRngElt] → SeqEnum, SeqEnum, SeqEnum`](decoding.md#function-cosetdecode-codelinrng-modtuprngelt-2)

    - [`Example: Spain Z4 12`](decoding.md#example-ex-68b44c)

  - [Syndrome Decoding](decoding.md#syndrome-decoding)

    - [`SyndromeDecode(C, u): CodeLinRng, ModTupFldElt → BoolElt, ModTupRngElt, ModTupFldElt`](decoding.md#function-syndromedecode-codelinrng-modtupfldelt)

    - [`SyndromeDecode(C, u): CodeLinRng, ModTupRngElt → BoolElt, ModTupRngElt, ModTupFldElt`](decoding.md#function-syndromedecode-codelinrng-modtuprngelt)

    - [`SyndromeDecode(C, Q): CodeLinRng, [ModTupFldElt] → SeqEnum, SeqEnum, SeqEnum`](decoding.md#function-syndromedecode-codelinrng-modtupfldelt-2)

    - [`SyndromeDecode(C, Q): CodeLinRng, [ModTupRngElt] → SeqEnum, SeqEnum, SeqEnum`](decoding.md#function-syndromedecode-codelinrng-modtuprngelt-2)

    - [`Example: Spain Z4 13`](decoding.md#example-ex-1a7dd2)

  - [Lifted Decoding](decoding.md#lifted-decoding)

    - [`LiftedDecode(C, u : parameters): CodeLinRng, ModTupFldElt → BoolElt, ModTupRngElt, ModTupFldElt`](decoding.md#function-lifteddecode-codelinrng-modtupfldelt)

    - [`LiftedDecode(C, u : parameters): CodeLinRng, ModTupRngElt → BoolElt, ModTupRngElt, ModTupFldElt`](decoding.md#function-lifteddecode-codelinrng-modtuprngelt)

    - [`LiftedDecode(C, Q : parameters): CodeLinRng, [ModTupFldElt] → SeqEnum, SeqEnum, SeqEnum`](decoding.md#function-lifteddecode-codelinrng-modtupfldelt-2)

    - [`LiftedDecode(C, Q : parameters): CodeLinRng, [ModTupRngElt] → SeqEnum, SeqEnum, SeqEnum`](decoding.md#function-lifteddecode-codelinrng-modtuprngelt-2)

    - [`Example: Hadamard Decoding`](decoding.md#example-ex-403c77)

  - [Permutation Decoding](decoding.md#permutation-decoding)

    - [`IsPermutationDecodeSet(C, I, S, s): CodeLinRng, [RngIntElt], [GrpPermElt], RngIntElt → BoolElt`](decoding.md#function-ispermutationdecodeset-codelinrng-rngintelt-grppermelt-rngintelt)

    - [`PermutationDecode(C, I, S, s, u): CodeLinRng, [RngIntElt], [GrpPermElt], RngIntElt, ModTupFldElt → BoolElt, ModTupRngElt, ModTupFldElt`](decoding.md#function-permutationdecode-codelinrng-rngintelt-grppermelt-rngintelt-modtupfldelt)

    - [`PermutationDecode(C, I, S, s, u): CodeLinRng, [RngIntElt], [GrpPermElt], RngIntElt, ModTupRngElt → BoolElt, ModTupRngElt, ModTupFldElt`](decoding.md#function-permutationdecode-codelinrng-rngintelt-grppermelt-rngintelt-modtuprngelt)

    - [`PermutationDecode(C, I, S, s, Q): CodeLinRng, [RngIntElt], [GrpPermElt], RngIntElt, [ModTupFldElt] → [BoolElt], [ModTupRngElt], [ModTupFldElt]`](decoding.md#function-permutationdecode-codelinrng-rngintelt-grppermelt-rngintelt-modtupfldelt-2)

    - [`PermutationDecode(C, I, S, s, Q): CodeLinRng, [RngIntElt], [GrpPermElt], RngIntElt, [ModTupRngElt] → [BoolElt], [ModTupRngElt], [ModTupFldElt]`](decoding.md#function-permutationdecode-codelinrng-rngintelt-grppermelt-rngintelt-modtuprngelt-2)

    - [`Example: Spain Z4 15`](decoding.md#example-ex-446a9b)

    - [`PDSetHadamardCodeZ4(δ, m): RngIntElt, RngIntElt → SeqEnum, SeqEnum, SeqEnum, SeqEnum, SeqEnum`](decoding.md#function-pdsethadamardcodez4-rngintelt-rngintelt)

    - [`PDSetKerdockCodeZ4(m): RngIntElt → SeqEnum, SeqEnum, SeqEnum, SeqEnum`](decoding.md#function-pdsetkerdockcodez4-rngintelt)

    - [`Example: Spain Z4 16`](decoding.md#example-ex-a6009d)

- [Automorphism Groups](aut_group.md)

  - [`PermutationGroupHadamardCodeZ4(δ, m): RngIntElt, RngIntElt → GrpPerm, Mtrx`](aut_group.md#function-permutationgrouphadamardcodez4-rngintelt-rngintelt)

  - [`PAutHadamardCodeZ4(δ, m): RngIntElt, RngIntElt → GrpPerm, Mtrx`](aut_group.md#function-pauthadamardcodez4-rngintelt-rngintelt)

  - [`PermutationGroupHadamardCodeZ4Order(δ, m): RngIntElt, RngIntElt → RngIntElt`](aut_group.md#function-permutationgrouphadamardcodez4order-rngintelt-rngintelt)

  - [`PAutHadamardCodeZ4Order(δ, m): RngIntElt, RngIntElt → RngIntElt`](aut_group.md#function-pauthadamardcodez4order-rngintelt-rngintelt)

  - [`PermutationGroupExtendedPerfectCodeZ4(δ, m): RngIntElt, RngIntElt → GrpPerm, Mtrx`](aut_group.md#function-permutationgroupextendedperfectcodez4-rngintelt-rngintelt)

  - [`PAutExtendedPerfectCodeZ4(δ, m): RngIntElt, RngIntElt → GrpPerm, Mtrx`](aut_group.md#function-pautextendedperfectcodez4-rngintelt-rngintelt)

  - [`PermutationGroupExtendedPerfectCodeZ4Order(δ, m): RngIntElt, RngIntElt → RngIntElt`](aut_group.md#function-permutationgroupextendedperfectcodez4order-rngintelt-rngintelt)

  - [`PAutExtendedPerfectCodeZ4Order(δ, m): RngIntElt, RngIntElt → RngIntElt`](aut_group.md#function-pautextendedperfectcodez4order-rngintelt-rngintelt)

  - [`Example: Spain Z4 8`](aut_group.md#example-ex-61ad8c)

  - [`PermutationGroup(C): CodeLinRng → GrpPerm`](aut_group.md#function-permutationgroup-codelinrng)

  - [`PermutationGroupGrayMapImage(C): CodeLinRng → GrpPerm`](aut_group.md#function-permutationgroupgraymapimage-codelinrng)

  - [`Example: Code Z4 Perm Group`](aut_group.md#example-ex-326517)