Algebraic Coding Theory Over Finite Commutative Rings PDF
Preview Algebraic Coding Theory Over Finite Commutative Rings
SPRINGER BRIEFS IN MATHEMATICS Steven T. Dougherty Algebraic Coding Theory Over Finite Commutative Rings 123 SpringerBriefs in Mathematics Series Editors Nicola Bellomo Michele Benzi Palle Jorgensen Tatsien Li Roderick Melnik Otmar Scherzer Benjamin Steinberg Lothar Reichel Yuri Tschinkel George Yin Ping Zhang SpringerBriefsinMathematicsshowcasesexpositionsinallareasofmathematics andappliedmathematics.Manuscriptspresentingnewresultsorasinglenewresult inaclassicalfield,newfield,oranemergingtopic,applications,orbridgesbetween newresultsandalreadypublishedworks,areencouraged.Theseriesisintendedfor mathematicians and applied mathematicians. More information about this series at http://www.springer.com/series/10030 Steven T. Dougherty Algebraic Coding Theory Over Finite Commutative Rings 123 StevenT. Dougherty Department ofMathematics University of Scranton Scranton, PA USA ISSN 2191-8198 ISSN 2191-8201 (electronic) SpringerBriefs inMathematics ISBN978-3-319-59805-5 ISBN978-3-319-59806-2 (eBook) DOI 10.1007/978-3-319-59806-2 LibraryofCongressControlNumber:2017943819 MathematicsSubjectClassification(2010): 11T71,94B05 ©TheAuthor(s)2017 Thisworkissubjecttocopyright.AllrightsarereservedbythePublisher,whetherthewholeorpart of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission orinformationstorageandretrieval,electronicadaptation,computersoftware,orbysimilarordissimilar methodologynowknownorhereafterdeveloped. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publicationdoesnotimply,evenintheabsenceofaspecificstatement,thatsuchnamesareexemptfrom therelevantprotectivelawsandregulationsandthereforefreeforgeneraluse. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authorsortheeditorsgiveawarranty,expressorimplied,withrespecttothematerialcontainedhereinor for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictionalclaimsinpublishedmapsandinstitutionalaffiliations. Printedonacid-freepaper ThisSpringerimprintispublishedbySpringerNature TheregisteredcompanyisSpringerInternationalPublishingAG Theregisteredcompanyaddressis:Gewerbestrasse11,6330Cham,Switzerland For Kelly, Steve and Checco. Acknowledgements The author is grateful to Jessica Hollister, Meg Hudock, Josep Rifà and Mercè Villanueva for their helpful comments on an early version of the text. vii Contents 1 Introduction.... .... .... ..... .... .... .... .... .... ..... .... 1 1.1 History.... .... .... ..... .... .... .... .... .... ..... .... 1 1.2 Definitions and Notations... .... .... .... .... .... ..... .... 3 References.. .... .... .... ..... .... .... .... .... .... ..... .... 10 2 Ring Theory.... .... .... ..... .... .... .... .... .... ..... .... 13 2.1 Finite Commutative Rings .. .... .... .... .... .... ..... .... 13 2.2 Frobenius Rings. .... ..... .... .... .... .... .... ..... .... 15 2.3 Chinese Remainder Theorem .... .... .... .... .... ..... .... 18 2.4 Generators . .... .... ..... .... .... .... .... .... ..... .... 22 References.. .... .... .... ..... .... .... .... .... .... ..... .... 27 3 MacWilliams Relations... ..... .... .... .... .... .... ..... .... 29 3.1 Introduction to the MacWilliams Relations.. .... .... ..... .... 29 3.2 MacWilliams Relations for Codes Over Groups.. .... ..... .... 30 3.3 MacWilliams Relations for Codes Over Rings... .... ..... .... 34 3.4 A Practical Guide to the MacWilliams Relations . .... ..... .... 37 References.. .... .... .... ..... .... .... .... .... .... ..... .... 40 4 Families of Rings.... .... ..... .... .... .... .... .... ..... .... 41 4.1 Rings of Order 4 .... ..... .... .... .... .... .... ..... .... 41 4.2 Ranks and Kernels of Quaternary Codes ... .... .... ..... .... 46 4.3 X-rings.... .... .... ..... .... .... .... .... .... ..... .... 48 4.4 The Ring Rq;D... .... ..... .... .... .... .... .... ..... .... 52 4.5 Chain Rings and Principal Ideal Rings. .... .... .... ..... .... 54 4.6 Generalized Singleton Bound.... .... .... .... .... ..... .... 55 References.. .... .... .... ..... .... .... .... .... .... ..... .... 57 5 Self-dual Codes . .... .... ..... .... .... .... .... .... ..... .... 59 5.1 Self-dual Codes Over Frobenius Rings. .... .... .... ..... .... 59 5.2 Connections to Lattices..... .... .... .... .... .... ..... .... 64 ix x Contents 5.3 Connections to Binary Self-dual Codes .... .... .... ..... .... 67 5.4 Connections to Designs .... .... .... .... .... .... ..... .... 72 5.5 Linear Complementary Dual. .... .... .... .... .... ..... .... 77 References.. .... .... .... ..... .... .... .... .... .... ..... .... 79 6 Cyclic and Constacyclic Codes . .... .... .... .... .... ..... .... 83 6.1 Polycyclic Codes .... ..... .... .... .... .... .... ..... .... 83 6.2 Constacyclic Codes Over Formal Power Series Rings and Chain Rings. .... ..... .... .... .... .... .... ..... .... 86 6.3 Codes as Ideals in Group Rings.. .... .... .... .... ..... .... 93 6.4 Quasicyclic Codes ... ..... .... .... .... .... .... ..... .... 95 6.5 h-Cyclic Codes.. .... ..... .... .... .... .... .... ..... .... 98 References.. .... .... .... ..... .... .... .... .... .... ..... .... 99 Index .... .... .... .... .... ..... .... .... .... .... .... ..... .... 101 Chapter 1 Introduction Inthischapter,wegiveabriefintroductiontothehistoryofalgebraiccodingtheory andgivethebasicdefinitionsandnotationsnecessarytobeginastudyofthesubject. 1.1 History Codingtheoryaroseinthetwentiethcenturyasaprobleminengineeringconcern- ing the efficient transmission of information. Its study originated in the landmark papersbyShannon[14]andHamming[9].Specifically,thetheorywasdevelopedso thatelectronicinformationcouldbetransmittedandstoredwithouterror.Electronic information can generally be thought of as a series of ones and zeros. Therefore, codingtheory,fromthisperspective,waslargelydoneusingthebinaryfieldasthe alphabet.However,thealphabetwasquicklygeneralizedtofinitefields,atleastfor mathematicians,sincemanyoftheproofsandtechniqueswereidenticaltothebinary case viewed as the field with two elements. This type of coding theory remains a vitalpartofelectricalengineeringintermsofensuringeffectivecommunicationin telephones,computers,television,andtheinternet. Fromtheverybeginningofitsstudy,mathematiciansviewedcodingtheorynot onlyasanapplicationtoelectricalengineeringandcomputerscience,butalsoasa partofpuremathematics.Theywereinterestednotonlyinthefundamentalquestions of coding theory, but also into its connections with other areas of discrete mathe- matics.Earlyresultsconnectedcodestodesigns,lattices,andcombinatorics.These connectionsweregenerallymadewithcodeswherethealphabetwasafinitefield. Moreover,muchoftheearlyworkfrommathematiciansincodingtheorycameby applyingpreviouslyknownresultsfromlinearalgebra,finitegeometry,algebra,and combinatoricstothestudyofcodes.Sincetheinceptionofcodingtheoryin1948, therehasbeenaveryfruitfulinterchangefrompuremathematicstotheapplicationof ©TheAuthor(s)2017 1 S.T.Dougherty,AlgebraicCodingTheoryOverFiniteCommutativeRings, SpringerBriefsinMathematics,DOI10.1007/978-3-319-59806-2_1 2 1 Introduction codes.Asoftenhappensinappliedmathematics,interestingmathematicalquestions aroseintheapplicationwhichsparkedmathematicians’interestinthesubject. Duringthefirstfortyyearsofcodingtheory,thealphabetinquestionwasusuallya finitefield.Therewereafewpaperswrittenwherethealphabetwasaring,forexample Blake’searlypapers[3, 4].Itwasn’tuntilthe1990swhencodingtheoristsbeganto studycodesoverfiniteringsinearnest.Thisstudybeganwiththeunderstandingthat certain non-linear binary codes, which had some of the properties of linear codes were,infact,theimagesofcodesoverZ underanon-linearmap.Thisbreakthrough 4 camein[10, 11],however,Delsarte’sworkin[6],yearsbefore,mighthaveledthe codingtheorycommunitytotheseresultsearlier.Thispromptedanintensestudyof codesoverZ whichrapidlymovedintothestudyofcodeswherethealphabetwas 4 eitheroneofthethreeothercommutativeringsoforder4ortheringZ .Familiesof k ringspresentedthemselvesforstudyandalargeliteratureemergedstudyingcodes overrings.Thefamiliesofringswereusuallychosenforsomespecificapplication. Forexample,codesoverthefamilyofringsZ werestudiedbecauseofaninteresting 2k connectiontounimodularreallattices,see[2].Itwasanaturalgeneralizationfrom thisfamilytothefamilyofchainrings.Later,withtheunderstandingthatallfinite commutativeringswerethedirectproductoflocalringsviatheChineseRemainder Theorem,codesoverlocalringswerestudied. In [16], J. Wood showed that both MacWilliams theorems held for the class of Frobenius rings. This showed that coding theory can be studied over this fairly largefamilyofringswithoutlosingthefundamentalfoundationsofcodingtheory. Generally, when studying codes over rings, a blanket assumption is made that all rings serving as alphabets for codes are finite Frobenius rings. An extensive and expandingliteraturenowexistsoncodesovervariousfamiliesofrings. Inthisbook,weshallnotdescribecodingtheoryasabranchofengineering,nor shallwemotivateitsstudyintermsofcommunicationapplications.Rather,weview codingtheoryasabranchofpuremathematicsservingasitsownmotivationforstudy. Weshallrefertothisbranchofpuremathematicsasalgebraiccodingtheory(whichit hasoftenalreadybeencalled)todistinguishitfromcodingtheoryasanapplicationin electricalengineering.Algebraiccodingtheorysitspartlyinalgebra,numbertheory, finitegeometry,andcombinatorics.Assuch,ithasinterestingconnectionstoawide varietyoftopicsinallthesebranches. The interested reader can consult MacWilliams and Sloane’s seminal text “The TheoryofError-CorrectingCodes”[13]foranearlydescriptionofclassicalcoding theory.Foranupdateddescription,seeHuffmanandPless’s“FundamentalsofError- correcting Codes” [12]. For a description of the connection between designs and codes see Assmus and Key’s “Designs and their Codes” [1]. In all three of these classictexts,codesaregenerallydefinedoverfinitefields. Inthistext,weshallbeconcernedwithcodesoverfinitecommutativeFrobenius ringsaswasfirstestablishedin[16].Itwillbecomeapparentwhyweneedtorestrict toFrobeniusringswhenwediscusstheMacWilliamsrelationsinChap.3.Weshall give foundational results for algebraic coding theory and develop the structures to view it as an interesting branch of pure mathematics. It is also possible to study codes over non-commutative rings, but much of the theory is different and as yet
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