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1、1YuanLuoXi’anJan.2013OptimumDistanceProfilesofLinearBlockCodesShanghaiJiaoTongUniversityHammingDistanceCodewordwithLongLengthorShortLengthOnewayis:Hammingdistance,generalizedHammingDistance,…AnotherDirectionis:Hammingdistance,distanceprofile,…Background:
2、OurresearchesonODPoflinearblockcode:Golay,RS,RM,cycliccodes,…3HammingDistance4Although,thelinearcodeswithlonglengtharemostoftenappliedinwirelesscommunication,thecodeswithshortlengthstillexistinindustry,forexample,somestoragesystems,theTFCIof3G(or4G)syste
3、m,somedatawithshortlengthbutneedstrongprotection,etc.CodewordwithLongLengthorShortLength5Forthecodeswithshortlength,thepreviousclassicboundscanhelpyoudirectly.Forthecodeswithlonglength,theasymptoticformsofthepreviousclassicboundsstillwork.Inthistopic,wec
4、onsidersomeproblemsinthefieldofHammingdistancewithshortcodewordlength.6Hammingdistanceisgeneralizedforthedescriptionoftrelliscomplexityoflinearblockcodes(DavidForney)andforthedescriptionofsecurityproblems(VictorWei).Wealsogeneralizedtheconcepttoconsidert
5、herelationshipbetweenacodeandasubcode:Onewayis:Hammingdistance,generalizedHammingDistance,…7Inthefollowing,weconsidertheHammingdistanceinavariationalsystem.Forexample,whentheencodinganddecodingdeviceswerealmostselected,butthetransmissionratedoesnotneedto
6、behighinaperiod(intheeveningnotsomuchusers),seenextslide,thenmoreredundanciescanbeborrowedtoimprovethedecodingability.Whatshouldwedotorealizethisidea?Andwhatistheprinciple?AnotherDirectionis:Hammingdistance,distanceprofile,…8TheTFCIin3Gsystem9DetailsInli
7、nearcodingtheory,whenthenumberofinputbitsincreasesordecreases,somebasiscodewordsofthegeneratormatrixwillbeincludedorexcluded,respectively.10Foragivenlinearblockcode,weconsider:※howtoselectageneratormatrixandthen※howtoincludeorexcludethebasiscodewordsofth
8、egeneratoronebyone※whilekeepingtheminimumdistances(ofthegeneratedsubcodes)aslargeaspossible.BigProblemIngeneralcase,thealgebraicstructuremaybelostinsubcodealthoughthepropertiesoftheoriginalcodearenice.Thenhowtodecode?1112O
