coding_theory

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1、IntroductiontoAlgebraicCodingTheoryWithGapFall2006SarahSpenceAdams¤January11,2008¤The¯rstversionsofthisbookwerewritteninFall2001andJune2002atCornellUniversity,respectivelysupportedbyanNSFVIGREGrantandaDepartmentofMathematicsGrant.Thecurrentversionwaswrittenin2005atF

2、ranklinW.OlinCollegeofEngineeringsupportedbyNSFCCLIDUE-0410517Grant.Editsweremadein2006underthesamegrant.Ifyouwishtousethisbook,pleasesimplyletmeknowandkeepmeinformedofanytyposorotherproblemsyou¯ndwiththebook.1Contents1IntroductiontoCodingTheory51.1IntroductoryExamp

3、les...................................61.2ImportantCodeParameters................................91.3CorrectingandDetectingErrors.............................111.4Sphere-PackingBound...................................131.5AnswerstoChapter1ReadingQuestions...........

4、.............151.6Chapter1Problems....................................162LinearCodes172.1BinaryLinearCodes....................................172.2Fields,VectorSpaces,andGeneralLinearCodes....................182.3EncodingLinearCodes:GeneratorMatrices................

5、.......222.4IntroductiontoParityCheckMatrices..........................252.5ParityCheckMatricesandLinearDecoding.......................282.6ParityCheckMatrices,MinimumDistance,andtheSingletonBound.........312.7HammingCodes......................................322.8A

6、nswerstoChapter2ReadingQuestions........................342.9Chapter2Problems....................................363CyclicCodes,Rings,andIdeals383.1IntroductiontoCyclicCodes...............................393.2RingsandIdeals......................................403.3

7、IdealsandCyclicCodes..................................433.4GeneratorandParityCheckPolynomials........................463.5AnswerstoChapter3ReadingQuestions........................493.6Chapter3Problems....................................494ClassesofPowerfulCyclicCode

8、s504.1SpecialCasesofBCHandRSCodes...........................514.2MinimalPolynomials....................................534.3BCHandReed-Sol