简介：Thispaperisdevotedtodeterminingthestructuresandpropertiesofone-Leeweightcodesandtwo-LeeweightprojectivecodesCk1,k2,k3overpIF+vIFpwithtypep2k1pk2pk3.Theauthorsintroduceadistance-preservingGraymapfrom(IFp+vIFp)nto2np.BytheGraymap,theauthorsconstructafamilyofoptimalone-Hammingweightp-arylinearcodesfromone-LeeweightcodesoverIFp+vIFp,whichattainthePlotkinboundandtheGriesmerbound.Theauthorsalsoobtainaclassofoptimalp-arylinearcodesfromtwo-LeeweightprojectivecodesoverIFp+vIFp,whichmeettheGriesmerbound.

简介：庇护重量枚举符，完全的重量在戒指R=F上为线性代码枚举符2+uF2+vF2被定义并且从R的灰色的地图n到F23n被构造。由证明灰色在R上自我双的代码想象的事实是在F上的自我双的代码2,并且基于为Hamming的MacWilliams身份，重量在F上线性代码枚举符2,MacWilliams为李重量的身份在R上线性代码枚举符被给。进一步由介绍特殊可变t，为完全的重量的MacWilliams身份在R上线性代码枚举符被获得。最后，说明二MacWilliams身份的正确性和功能的一个例子被提供。

简介：Weinthispaperexaminewarrantystrategyinatwo-stagesupplychainconsistingofamanufacturerandtwocompetingretailers.Themanufacturerproducestwosubstituteproductsandmarketsthemthroughthetworetailerstoagroupofconsumers,respectively.Foreachtypeofproducts,themanufacturer’sbasewarrantyandaretailer’sextendedwarrantyarebundledwiththeproduct.Weusegametheoreticmodelstoexploretheinteractionsbetweenthetwotypesofwarrantiesandthecompetitionbetweentheretailers.Forthispurpose,twoscenariosareconsidered:noretailerandboththetworetailersprovidingtheextendedwarranties,respectively.Ineachscenario,themanufacturer’sbasewarrantiesareassumedtobeoffered.Ourresultsshowthatwhentheretailersoffertheirextendedwarranties,themanufacturerhasnoincentivetoofferthebasewarranties;otherwise,themanufacturerhastoprovidethebasewarranties.Thecompetitionbetweenretailersintermsoftheproductsubstitutabilityhasnoimpactonwarrantydecisions,butaffectsallplayers’profitsinthesupplychain.Themanufacturercanprovidealongerwarrantylengthandhighercustomerwelfaretoacustomerthantheretailersdo,ifitismoreefficientthantheretailersinwarrantycost-efficiency,andviceversa.