Linear Codes over the Integer Residue Ring \({\mathbb{Z}}_4\)
- Introduction
- Constructions for \({\mathbb{Z}}_4\) Codes
- The Gray Map
- Families of Codes over \({\mathbb{Z}}_4\)
KerdockCode(m): RngIntElt → CodePreparataCode(m): RngIntElt → CodeReedMullerCodeZ4(r, m): RngIntElt, RngIntElt → CodeGoethalsCode(m): RngIntElt → CodeDelsarteGoethalsCode(m, delta): RngIntElt, RngIntElt → CodeGoethalsDelsarteCode(m, delta): RngIntElt, RngIntElt → CodeQRCodeZ4(p): RngIntElt → CodeGolayCodeZ4(e): BoolElt → CodeSimplexAlphaCodeZ4(k): RngIntElt → CodeSimplexBetaCodeZ4(k): RngIntElt → CodeExample: KerdockHadamardCodeZ4(δ, m): RngIntElt, RngIntElt → CodeLinRng, MtrxExtendedPerfectCodeZ4(δ, m): RngIntElt, RngIntElt → CodeLinRng, MtrxExample: Spain Z4 1ReedMullerCodeZ4(r, m): RngIntElt, RngIntElt → CodeLinRngReedMullerCodeQRMZ4(r, m): RngIntElt, RngIntElt → CodeLinRngReedMullerCodesLRMZ4(r, m): RngIntElt, RngIntElt → SeqEnumReedMullerCodeRMZ4(s, r, m): RngIntElt, RngIntElt, RngIntElt → CodeLinRng, MtrxExample: Spain Z4 2ReedMullerCodesRMZ4(s, m): RngIntElt, RngIntElt → TupExample: Spain Z4 3
- Derived Binary Codes
- New Codes from Old
PlotkinSum(A, B): Mtrx, Mtrx → MtrxPlotkinSum(C, D): Code, Code → CodeQuaternaryPlotkinSum(A, B): Mtrx, Mtrx → MtrxQuaternaryPlotkinSum(C, D): Code, Code → CodeBQPlotkinSum(A, B, C): Mtrx, Mtrx, Mtrx → MtrxBQPlotkinSum(D, E, F): Code, Code, Code → CodeDoublePlotkinSum(A, B, C, D): Mtrx, Mtrx, Mtrx, Mtrx → MtrxDoublePlotkinSum(E, F, G, H): Code, Code, Code, Code → CodeDualKroneckerZ4(C): CodeLinRng → CodeLinRngExample: Spain Z4 4Example: Spain Z4 4a
- Invariants
- Weight Distributions
- Weight Enumerators
- Decoding
- Coset Decoding
CosetDecode(C, u : parameters): CodeLinRng, ModTupFldElt → BoolElt, ModTupRngElt, ModTupFldEltCosetDecode(C, u : parameters): CodeLinRng, ModTupRngElt → BoolElt, ModTupRngElt, ModTupFldEltCosetDecode(C, Q : parameters): CodeLinRng, [ModTupFldElt] → SeqEnum, SeqEnum, SeqEnumCosetDecode(C, Q : parameters): CodeLinRng, [ModTupRngElt] → SeqEnum, SeqEnum, SeqEnumExample: Spain Z4 12
- Syndrome Decoding
SyndromeDecode(C, u): CodeLinRng, ModTupFldElt → BoolElt, ModTupRngElt, ModTupFldEltSyndromeDecode(C, u): CodeLinRng, ModTupRngElt → BoolElt, ModTupRngElt, ModTupFldEltSyndromeDecode(C, Q): CodeLinRng, [ModTupFldElt] → SeqEnum, SeqEnum, SeqEnumSyndromeDecode(C, Q): CodeLinRng, [ModTupRngElt] → SeqEnum, SeqEnum, SeqEnumExample: Spain Z4 13
- Lifted Decoding
LiftedDecode(C, u : parameters): CodeLinRng, ModTupFldElt → BoolElt, ModTupRngElt, ModTupFldEltLiftedDecode(C, u : parameters): CodeLinRng, ModTupRngElt → BoolElt, ModTupRngElt, ModTupFldEltLiftedDecode(C, Q : parameters): CodeLinRng, [ModTupFldElt] → SeqEnum, SeqEnum, SeqEnumLiftedDecode(C, Q : parameters): CodeLinRng, [ModTupRngElt] → SeqEnum, SeqEnum, SeqEnumExample: Hadamard Decoding
- Permutation Decoding
IsPermutationDecodeSet(C, I, S, s): CodeLinRng, [RngIntElt], [GrpPermElt], RngIntElt → BoolEltPermutationDecode(C, I, S, s, u): CodeLinRng, [RngIntElt], [GrpPermElt], RngIntElt, ModTupFldElt → BoolElt, ModTupRngElt, ModTupFldEltPermutationDecode(C, I, S, s, u): CodeLinRng, [RngIntElt], [GrpPermElt], RngIntElt, ModTupRngElt → BoolElt, ModTupRngElt, ModTupFldEltPermutationDecode(C, I, S, s, Q): CodeLinRng, [RngIntElt], [GrpPermElt], RngIntElt, [ModTupFldElt] → [BoolElt], [ModTupRngElt], [ModTupFldElt]PermutationDecode(C, I, S, s, Q): CodeLinRng, [RngIntElt], [GrpPermElt], RngIntElt, [ModTupRngElt] → [BoolElt], [ModTupRngElt], [ModTupFldElt]Example: Spain Z4 15PDSetHadamardCodeZ4(δ, m): RngIntElt, RngIntElt → SeqEnum, SeqEnum, SeqEnum, SeqEnum, SeqEnumPDSetKerdockCodeZ4(m): RngIntElt → SeqEnum, SeqEnum, SeqEnum, SeqEnumExample: Spain Z4 16
- Automorphism Groups
PermutationGroupHadamardCodeZ4(δ, m): RngIntElt, RngIntElt → GrpPerm, MtrxPAutHadamardCodeZ4(δ, m): RngIntElt, RngIntElt → GrpPerm, MtrxPermutationGroupHadamardCodeZ4Order(δ, m): RngIntElt, RngIntElt → RngIntEltPAutHadamardCodeZ4Order(δ, m): RngIntElt, RngIntElt → RngIntEltPermutationGroupExtendedPerfectCodeZ4(δ, m): RngIntElt, RngIntElt → GrpPerm, MtrxPAutExtendedPerfectCodeZ4(δ, m): RngIntElt, RngIntElt → GrpPerm, MtrxPermutationGroupExtendedPerfectCodeZ4Order(δ, m): RngIntElt, RngIntElt → RngIntEltPAutExtendedPerfectCodeZ4Order(δ, m): RngIntElt, RngIntElt → RngIntEltExample: Spain Z4 8PermutationGroup(C): CodeLinRng → GrpPermPermutationGroupGrayMapImage(C): CodeLinRng → GrpPermExample: Code Z4 Perm Group
