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 → Code- Example: Kerdock and Preparata Codes
HadamardCodeZ4(δ, m): RngIntElt, RngIntElt → CodeLinRng, MtrxExtendedPerfectCodeZ4(δ, m): RngIntElt, RngIntElt → CodeLinRng, Mtrx- Example: Hadamard Codes over Z4
ReedMullerCodeZ4(r, m): RngIntElt, RngIntElt → CodeLinRngReedMullerCodeQRMZ4(r, m): RngIntElt, RngIntElt → CodeLinRngReedMullerCodesLRMZ4(r, m): RngIntElt, RngIntElt → SeqEnumReedMullerCodeRMZ4(s, r, m): RngIntElt, RngIntElt, RngIntElt → CodeLinRng, Mtrx- Example: Nested Reed-Muller Codes over Z4
ReedMullerCodesRMZ4(s, m): RngIntElt, RngIntElt → Tup- Example: Family of Reed-Muller Codes over Z4
- 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 → CodeLinRng- Example: Plotkin Sums of Matrices and Codes
- Example: Reed-Muller Codes as Plotkin Sums
- 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, SeqEnum- Example: Coset Decoding of a Hadamard Code
- 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, SeqEnum- Example: Syndrome Decoding of a Hadamard Code
- 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, SeqEnum- Example: Lifted Decoding of a Hadamard Code
- 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: Permutation Decoding of a Hadamard Code
PDSetHadamardCodeZ4(δ, m): RngIntElt, RngIntElt → SeqEnum, SeqEnum, SeqEnum, SeqEnum, SeqEnumPDSetKerdockCodeZ4(m): RngIntElt → SeqEnum, SeqEnum, SeqEnum, SeqEnum- Example: Checking a PD-Set for a Hadamard Code
- 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 → RngIntElt- Example: Permutation Automorphisms of a Hadamard Code
PermutationGroup(C): CodeLinRng → GrpPermPermutationGroupGrayMapImage(C): CodeLinRng → GrpPerm- Example: Permutation Group of a Code over Z4
- Bibliography