invariance of generalized wordlength patternsnew

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1、InvarianceofgeneralizedwordlengthpatternsJayH.BederJebF.WillenbringDepartmentofMathematicalSciencesUniversityofWisconsin-MilwaukeeP.O.Box413Milwaukee,WI53201-0413beder@uwm.edu,jw@uwm.eduAbstractThegeneralizedwordlengthpattern(GWLP)introducedbyXuandWu(2001)foranarbitraryfractionalfactoria

2、ldesignallowsonetoextendtheuseoftheminimumaber-rationcriteriontosuchdesigns.AiandZhang(2004)de¯nedtheJ-characteristicsofadesignandshowedthattheyuniquelydeterminethedesign.WhileboththeGWLPandtheJ-characteristicsrequireindexingthelevelsofeachfactorbyacyclicgroup,weseethatthede¯nitionscarry

3、overwithappropriatechangesifinsteadoneusesanarbitraryabeliangroup.Thismeansthattheoriginalde¯nitionsrestonanarbitrarychoiceofgroupstructure.WeshowthattheGWLPofadesignisindependentofthischoice,butthattheJ-characteristicsarenot.Webrie°ydiscusssomeimplicationsoftheseresults.Keywords.Fractio

4、nalfactorialdesign;groupcharacter;Hammingweight;multiset;orthogonalarrayAMS(MOS)subjectclassi¯cation.Primary:62K15;Secondary:05B15,15A69,20C15,62K051IntroductionInaregularfractionalfactorialdesignD,thequantitiesAi(D)=thenumberofde¯ningwordsoflengthicontainusefulinformationaboutthedesign.

5、Inparticular,thesmallestindexiforwhichAi(D)>0istheresolutionofthedesign.Moreover,onewayofcomparingtwodesignshavingkfactorsandequalresolutionistocomparetheirenterprisepatterns(A1;A2;:::;Ak)[5,6].Thebetterdesignissaidtohavelessaberration.Whilenonregulardesignsnolongerhavede¯ningwordsassuch

6、,ageneralizedwordlengthpattern(GWLP)canbede¯nedforthemcombinatorially.Thiswasdonefortwo-leveldesignsbyTangandDeng[12],andwasgeneralizedtoarbitrary(possiblymixed-level)designsbyXuandWu[14]usinggroupcharacters.Anintermediatecomputationinthetwo-levelcasegivesasetofvaluesthatTangandDengcalle

7、dJ-characteristics(¯rstintroducedin[4]),andTang[11]showedthatthesenumberscompletelydeterminethedesignD,somewhatanalogoustothewaythatade¯ningsubgroupdeterminesaregulardesign.AiandZhang[1]generalizedthistoarbitrarydesignsbylookingcloselyatthecorrespondingcomputationin[14].1