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1、1FromRandomMatrixTheorytoCodingTheory:VolumeofaMetricBallinUnitaryGroupLuWei,Renaud-AlexandrePitaval,JukkaCorander,andOlavTirkkonenAbstractVolumeestimatesofmetricballsinmanifolds?nddiverseapplicationsininformationandcodingtheory.Inthispaper,somenewresultsforthevolumeofametricballinunitarygroupared
2、erivedviavarioustoolsfromrandommatrixtheory.The?rstresultisanintegralrepresentationoftheexactvolume,whichinvolvesaToeplitzdeterminantofBesselfunctions.Theconnectiontomatrix-variatehypergeometricfunctionsandSzego’sstronglimittheoremleadindependentlyfromthe?nitesizeformula?toanasymptoticone.Theconve
3、rgenceofthelimitingformulaisexceptionallyfastduetoanunderlyingmock-Gaussianbehavior.Theproposedvolumeestimateenablessimplebutaccurateanalyticalevaluationofcoding-theoreticboundsofunitarycodes.Inparticular,theGilbert-VarshamovlowerboundandtheHammingupperboundoncardinalityaswellastheresultingboundso
4、ncoderateandminimumdistancearederived.Moreover,boundsonthescalinglawofcoderatearefound.Lastly,aclosed-formboundondiversitysumrelevanttounitaryspace-timecodesisobtained,whichwasonlycomputednumericallyinliterature.arXiv:1506.07259v1[cs.IT]24Jun2015IndexTermsCoding-theoreticbounds,randommatrixtheory,
5、unitarygroup,volumeofmetricballs.L.WeiandJ.CoranderarewiththeDepartmentofMathematicsandStatistics,UniversityofHelsinki,Finland(e-mails:{lu.wei,jukka.corander}@helsinki.?).R.-A.PitavaliswiththeDepartmentofMathematicsandSystemsAnalysis,AaltoUniversity,Finland(e-mail:renaud-alexandre.pitaval@aalto.?)
6、.O.TirkkoneniswiththeDepartmentofCommunicationsandNetworking,AaltoUniversity,Finland(e-mail:olav.tirkkonen@aalto.?).Thisworkwaspresentedinpartat2015IEEEInternationalSymposiumonInformationTheory.June25,2015DRAFT2I.INTRODUCTIONDeterminingthevolumeofmetricballsinRiemannianmanifold,inparticularunitary
7、group,isthekeytounderstandseveralcodingandinformationtheoreticalquantities.Performanceanalysisofunitaryspace-timecodes[1–3]requirestheknowledgeofvolumeintheunitarygroup[4,5].Forchannelquantizationsinprecodedmulti
