Network Coding and Subspace Designs PDF
Preview Network Coding and Subspace Designs
Signals and Communication Technology Marcus Greferath Mario Osvin Pavčević Natalia Silberstein María Ángeles Vázquez-Castro E ditors Network Coding and Subspace Designs Signals and Communication Technology More information about this series at http://www.springer.com/series/4748 č ć Marcus Greferath Mario Osvin Pav evi (cid:129) í Á á Natalia Silberstein Mar a ngeles V zquez-Castro (cid:129) Editors Network Coding and Subspace Designs 123 Editors Marcus Greferath Natalia Silberstein Department ofMathematics Yahoo Research andSystemsAnalysis Haifa AaltoUniversity Israel Espoo Finland MaríaÁngeles Vázquez-Castro Department ofTelecommunications Mario OsvinPavčević andSystemsEngineering Faculty of Electrical Engineering Universitat Autònoma deBarcelona andComputing,Department Cerdanyola del Vallès,Barcelona ofApplied Mathematics Spain University of Zagreb Zagreb Croatia ISSN 1860-4862 ISSN 1860-4870 (electronic) Signals andCommunication Technology ISBN978-3-319-70292-6 ISBN978-3-319-70293-3 (eBook) https://doi.org/10.1007/978-3-319-70293-3 LibraryofCongressControlNumber:2017958828 ©SpringerInternationalPublishingAG2018 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. 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Printedonacid-freepaper ThisSpringerimprintispublishedbySpringerNature TheregisteredcompanyisSpringerInternationalPublishingAG Theregisteredcompanyaddressis:Gewerbestrasse11,6330Cham,Switzerland This article/publication is based upon work from COST Action IC1104 Random Network Coding and Designs over GF(q), supported by COST (European CooperationinScienceandTechnology). COST (European Cooperation in Science and Technology) is a funding agency forresearchandinnovationnetworks.OurActionshelpconnectresearchinitiatives across Europe and enable scientists to grow their ideas by sharing them with their peers. This boosts their research, career and innovation. http://www.cost.eu FundedbytheHorizon2020FrameworkProgramme oftheEuropeanUnion v Foreword The beautiful and the elegant are often inspired by the seemingly mundane. The chapters of this book follow from a line of work rooted in a practical engi- neering problem: namely, the efficient transmission of information in packet net- works.Traditionalapproachestotheoperationofpacketnetworkstreatinformation flowascommodityflow,emphasizingtheefficientroutingoftheinformationalong networkpathways,whileavoidingorcleverlyresolvingcontentionfortransmission resources. Network coding, introduced in 2000 in the seminal paper of Ahlswede, Cai, Li, and Yeung, challenges this conventional approach. Networkcodingisbasedonasimple,yetfar-reaching,idea:Ratherthansimply routing packets, intermediate nodes are permitted to “mix” packets, transmitting functions of the data that they receive. At the network boundary, received packets are treated as evidence that must be gathered in sufficient quantity so as to permit decoding of the transmitted message. A special case arises when the packets are interpreted as vectors of symbols drawn from a finite field, and the local mixing functions are chosen to be linear transformations. Fromthe2003paperofLi,Yeung,andCai,itisknownthatsuch linear network coding can achieve, when the underlying field is large enough, the so-called multicast capacity of a network, in which a single source wishes to communicate the same message to a number of different terminals. An elegant algebraicproofofthisfactisgiveninthe2003paperofKötterandMédardviathe existenceofanonzeroforaparticularmultivariatepolynomialarisingasaproduct of matrix determinants. Indeed, as was demonstrated in the 2006 paper of Ho, Médard,Kötter,Karger,Effros,Shi,andLeong,themulticastcapacityofanetwork is achieved, with high probability in a sufficiently large field, by a completely random choice of local mixing functions, obviating the need for a deliberate network-topology-dependent code design. Such random linear network coding yields an interesting new type of data transmissionmodel.Inthismodel,tosendamessagethetransmitterinjectsintothe network a collection of vectorswhich propagatethrough intermediate nodes inthe network, where they are randomly linearly combined, before arriving at any given terminal.Aterminalobservesanumberofsuchvectors,fromwhichthetransmitted vii viii Foreword message must be inferred. Since the linear transformation by the channel of the transmitted vectors is not known in advance by the transmitter or any of the receivers, such a model is sometimes referred to as a noncoherent transmission model (in contrast with a so-called coherent model, in which knowledge of the channel transformation is assumed). Toachieveinformationtransmissioninthisnoncoherentmodelofrandomlinear network coding, one might seek a communication invariant: some property of the transmittedvectorsthatispreservedbytheoperationofthechannel.Asobservedin my2008paperwithRalfKötter,anessentialinvariant—thekeypropertypreserved by the random action of the channel—is the vector space spanned by the trans- mitted vectors. No matter the action of the channel, we are guaranteed (in the absence of errors) that each of the vectors observed at any terminal belongs to the space spanned by the transmitted vectors. Thus, we are naturally led to consider information transmission not via the choice of the transmitted vectors themselves, but rather by the choice of the vector space that they span. Suchatransmissionmodelalsolendsitselftoaconcisedescriptionoftheeffects of the injection (by an adversary, say) of erroneous vectors into the network: The erroneous vectors combine linearly with the transmitted ones, resulting in a vector spaceobservedatanyreceiverthatmaydifferfromthetransmittedone.Whenevera spaceV istransmitted(byinjectionintothenetworkofabasisforV),andaspaceU is observed at a terminal (by observation of a basis for U), the transmitted V is related tothereceived U viathedirectsum U ¼HðVÞ(cid:2)E, where theoperatorH selectssomesubspaceofV andEisasuitableerrorspaceintersectingtriviallywith V.Thisso-calledoperatorchanneltakesinavectorspacefromthetransmitterand produces some vector space at any given receiver, where the received space may suffer from erasures (deletion of vectors from the transmitted space) or errors (addition of vectors to the transmitted space). Acodingtheoryfortheoperatorchannelthuspresentsitselfverynaturally.The collectionPðWÞofsubspacesoftheambientpacketspaceW playstheroleofinput andoutput alphabets.One may define metrics onPðWÞtomeasuretheadversarial effortrequiredtoconvertatransmittedsubspaceV toareceivedoneU;intuitively, two spaces shouldbe near eachother if they intersect ina space ofrelativelylarge dimension. One natural measure, which equally weights erasures and errors, is the “subspace metric” d ðU;VÞ¼dimðUþVÞ(cid:3)dimðU\WÞ. Another measure, S introduced in my 2008 paper with Silva and Kötter, is the “injection distance” d ðU;VÞ¼maxfdimðUÞ;dimðVÞg(cid:3)dimðU\VÞ, which accounts for the possi- I bility that a single injection of a packet by an adversary may simultaneously case the deletion of a dimension (an erasure) and the insertion of one (an error). These twometricscoincide(exceptforafactoroftwo)inthecaseofconstantdimension, i.e., in the case when dimðUÞ¼dimðVÞ. A subspace code for the operator channel is then a nonempty subset of PðWÞ, i.e.,anonemptycodebookofvectorspaces,eachasubspaceoftheambientpacket space W.Asinclassical coding theory,theerror- anderasure-correcting capability of such a code is determined by the minimum distance between codewords, mea- sured according to either the subspace distance or the injection distance. In the Foreword ix importantspecialcasewhenthecodewordsallhavethesamedimension,aso-called constantdimensioncode,ananalogofa“constantweightcode”inclassicalcoding theory, arises. Many very interesting mathematical and engineering questions arise immedi- ately; indeed, the chapters of this book aim to pose, study, and answer some of them. As in classical coding theory, one can ask for extremal codes. For example, when the ambient packet space is Fn, which codes with codewords of constant q dimension k have maximum codebook cardinality M while preserving minimum injectiondistanced?Similarly,forafixedcodebookcardinalityM,whichcodesof constant dimension k have largest possible minimum injection distance d? One might also be interested in extremal codes with codewords not all of the same dimension, in which case the distinction between the subspace distance and the injection distance becomes important. Many classical coding theory bounds on extremalcodes(e.g.,theHammingbound,theSingletonbound,theGilbertbound) have analogs in the new setting of subspace codes. Onemayalsoaskforcodeconstructionsthatadmitefficientdecodingalgorithms. Inthecasewherek jn,theanalogofaclassicalextremalcode,therepetitioncode, is now an extremal object in finite geometry: a spread, a collection of pairwise triviallyintersectingk-dimensionalsubspacesofFn whoseunionisFn.Inour2008 q q papers,weshowedthatconstantdimensioncodesofverylarge(“nearly”extremal) cardinality can be constructed by a “lifting” of extremal codes—such as the maxi- mum rank-distance Delsarte–Gabidulin codes—constructed for the rank metric. Thesecodesconsistoftherowspacesofmatricesoftheform½I jM(cid:4),whereI isthe identitymatrixandM isamatrixcodewordofarank-metriccode.Furthermore,we showedthatefficientanalogsofclassicalboundeddistancedecodingalgorithmsfor Reed–Solomon codes canbedeveloped for such lifted Delsarte–Gabidulincodes. Closely related to the combinatorial questions of code construction, a large part of combinatorial mathematics and finite geometry deals with the existence, construction, and properties of so-called combinatorial designs: collections of sub- sets of some ambient set exhibiting particular balance or symmetry properties. AprototypicalexampleisaSteinersystem:acollectionofk-elementsubsets(called blocks) of S¼f1;...;ng with the property that each t-element subset of S is con- tainedinexactlyoneblock.Inthiscontextofsubspacesofafixedambientspace,one can ask for q-analogs of such designs: namely, collections of subspaces of some ambient space over F exhibiting particular balance or symmetry properties. For q example, the q-analog of a Steiner system would be a collection of k-dimensional subspaces(blocks)ofW ¼Fnwiththepropertythateacht-dimensionalsubspaceof q W iscontainedinexactlyoneblock.Thatnontrivial(t(cid:5)2)q-Steinersystemseven exist was not known until the 2013 paper of Braun, Etzion, Östergård, Vardy, and Wassermann. Written by leading experts in the field, this book is an exploration of the beautifulandelegantmathematicalandengineeringideasthatarerelatedtonetwork coding.Here,youwillfindnewconstructionsofsubspacecodesofvarioustypesas x Foreword well as interesting generalizations. You will find a deep discussion of rank-metric codes and their properties, and you will find connections made to finite geometry and the theory of combinatorial designs, including a description of state-of-the-art computational methods that can be used to search for designs. You will learn how network coding is related to problems of broadcasting with side information at the receivers, and how network coding can be applied in various wireless communi- cationscenarios.Networkcodingideashavealsobeenusedtodetermineboundson theparametersofcodesthatareusefulindistributedstoragesystems.Inthisbook, you will find a nice overview of the desirable features—such as “locality”—that such codes should possess, along with new constructions for locally repairable codes, and codes that permit so-called private information retrieval from a dis- tributed storage system. That such a diversity of new ideas should arise from the problem of efficient transmission ofinformationinapacket networksuggeststhattheoriginalproblem may not have been so mundane after all. Toronto, Canada Frank R. Kschischang Dept. of Electrical & Computer Engineering University of Toronto
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