Coding Theory and Applications: Second International Castle Meeting, ICMCTA 2008, Castillo de la Mota, Medina del Campo, Spain, September 15-19, 2008. Proceedings PDF
Preview Coding Theory and Applications: Second International Castle Meeting, ICMCTA 2008, Castillo de la Mota, Medina del Campo, Spain, September 15-19, 2008. Proceedings
Lecture Notes in Computer Science 5228 CommencedPublicationin1973 FoundingandFormerSeriesEditors: GerhardGoos,JurisHartmanis,andJanvanLeeuwen EditorialBoard DavidHutchison LancasterUniversity,UK TakeoKanade CarnegieMellonUniversity,Pittsburgh,PA,USA JosefKittler UniversityofSurrey,Guildford,UK JonM.Kleinberg CornellUniversity,Ithaca,NY,USA AlfredKobsa UniversityofCalifornia,Irvine,CA,USA FriedemannMattern ETHZurich,Switzerland JohnC.Mitchell StanfordUniversity,CA,USA MoniNaor WeizmannInstituteofScience,Rehovot,Israel OscarNierstrasz UniversityofBern,Switzerland C.PanduRangan IndianInstituteofTechnology,Madras,India BernhardSteffen UniversityofDortmund,Germany MadhuSudan MassachusettsInstituteofTechnology,MA,USA DemetriTerzopoulos UniversityofCalifornia,LosAngeles,CA,USA DougTygar UniversityofCalifornia,Berkeley,CA,USA GerhardWeikum Max-PlanckInstituteofComputerScience,Saarbruecken,Germany Ángela Barbero (Ed.) Coding Theory and Applications Second International Castle Meeting, ICMCTA 2008 Castillo de la Mota, Medina del Campo, Spain September 15-19, 2008 Proceedings 1 3 VolumeEditor ÁngelaBarbero DepartamentodeMatemáticaAplicada(U.Valladolid) E.T.S.IngenierosIndustriales PaseodelCauces/n,47011 Valladolid,Spain E-mail:[email protected] LibraryofCongressControlNumber:2008934288 CRSubjectClassification(1998):E.4,I.1,G.2,E.3 LNCSSublibrary:SL1–TheoreticalComputerScienceandGeneralIssues ISSN 0302-9743 ISBN-10 3-540-87447-XSpringerBerlinHeidelbergNewYork ISBN-13 978-3-540-87447-8SpringerBerlinHeidelbergNewYork Thisworkissubjecttocopyright.Allrightsarereserved,whetherthewholeorpartofthematerialis concerned,specificallytherightsoftranslation,reprinting,re-useofillustrations,recitation,broadcasting, reproductiononmicrofilmsorinanyotherway,andstorageindatabanks.Duplicationofthispublication orpartsthereofispermittedonlyundertheprovisionsoftheGermanCopyrightLawofSeptember9,1965, initscurrentversion,andpermissionforusemustalwaysbeobtainedfromSpringer.Violationsareliable toprosecutionundertheGermanCopyrightLaw. SpringerisapartofSpringerScience+BusinessMedia springer.com ©Springer-VerlagBerlinHeidelberg2008 PrintedinGermany Typesetting:Camera-readybyauthor,dataconversionbyScientificPublishingServices,Chennai,India Printedonacid-freepaper SPIN:12519927 06/3180 543210 Preface It is a pleasure to welcome you to the proceedings of the second International Castle Meeting on Coding Theory and its Applications, held at La Mota Castle in Medina del Campo. The event provided a forum for the exchange of results and ideas, which we hope will foster future collaboration.The first meeting was held in 1999, and, encouraged by that experience, we now intend to hold the meeting every three years. Springerkindlyacceptedtopublishtheproceedingsvolumeyouhaveinyour hands in their LNCS series. The topics were selected to cover some of the areas of research in Coding Theory that are currently receiving the most attention. The program consisted of a mixture of invited and submitted talks, with the focus on quality rather than quantity. A total of 34 papers were submitted to themeeting.Afteracarefulreviewprocessconductedbythescientificcommittee aided by external reviewers, we selected 14 of these for inclusion in the current volume, along with 5 invited papers. The program was further augmented by the remaining invited papers in addition to papers on recent results, printed in a separate volume. We wouldlike to thank everyonewho made this meeting possible by helping with the practical and scientific preparations: the organization committee, the scientific committee, the invited speakers,and the many externalreviewerswho shallremainanonymous.I wouldespeciallylike tomentionthe GeneralAdvisor ofthemeeting,ØyvindYtrehus.FinallyIextendmygratitudetoalltheauthors and participants who contributed to this meeting. We thank the official institutions that sponsored our efforts: the Proyecto Consolider “Ingenio Mathematica” and the University of Valladolid. This preface would not be complete without a few words about the special environmentprovidedby the castle that gaveits name to the meeting. La Mota Castle was built on the remains of a moorish castle from the 12th century. The currentarchitectureisfromthe15thcentury.Inthosedays,thecastlewasoneof thebiggestandmostimportantcastlesinEurope,locatedinMedinadelCampo, one of the main trade centers of the Spanish Empire. The castle has lodged historical characters like Queen Isabel the Catholic and her daughter Joan the Mad,anditsdungeonshaveprovidedaccommodationforotherprominentguests like C´esar Borgia (who managed to escape by use of a file and a rope). We are sure such an atmosphere will generate inspiration for future work on the topics of the meeting and in many other directions. July 2008 A´ngela Barbero Organization 2ICMCTA 2008 was organized by the University of Valladolid and was held in the Castillo de la Mota in Medina del Campo. Organizing Committee General Chair A´ngela Barbero (University of Valladolid, Spain) Co-chair Juan Tena (University of Valladolid, Spain) General Advisor Øyvind Ytrehus (University of Bergen, Norway) Other members M. Francisca Blanco (University of Valladolid, Spain) Javier Gala´n (University of Valladolid, Spain) Carlos Munuera (University of Valladolid, Spain) Daniel Sadornil (University of Cantabria, Spain) Scientific Committee Maria Bras-Amoro´s University Rovira i Virgili, Spain Gerard Cohen ENST Paris, France Tom Høholdt Denmark Technological University, Denmark Ignacio Luengo Complutense University of Madrid, Spain Garegin Markarian Lancaster University, UK Consuelo Mart´ınez University of Oviedo, Spain Matthew Parker University of Bergen, Norway Kevin Phelps Auburn University, USA Josep Rif´a Autonomous University of Barcelona, Spain Eirik Rosnes University of Bergen, Norway Emina Soljanin Bell Labs, USA Faina Solov’eva University of Novosibirsk,Russia Ludo Tolhuizen Philips Research, Eindhoven, The Netherlands Merc´e Villanueva Autonomous University of Barcelona, Spain Jos Weber Technical University of Delft, The Netherlands Øyvind Ytrehus University of Bergen, Norway Sponsoring Institutions Ingenio Mathematica (project Consolider) University of Valladolid Table of Contents A Diametric Theorem in Zn for Lee and Related Distances ........... 1 m Rudolf Ahlswede and Faina I. Solov’eva Perfect 2-Colorings of Johnson Graphs J(6,3) and J(7,3) ............. 11 Sergey Avgustinovich and Ivan Mogilnykh A Syndrome Formulation of the Interpolation Step in the Guruswami-Sudan Algorithm...................................... 20 Peter Beelen and Tom Høholdt How to Know if a Linear Code Is a Group Code? .................... 33 Jos´e Joaqu´ın Bernal, A´ngel del R´ıo, and Juan Jacobo Sim´on Witness Sets .................................................... 37 G´erard Cohen, Hugues Randriam, and Gilles Z´emor On Rank and Kernel of Z -Linear Codes............................ 46 4 Cristina Ferna´ndez-C´ordoba, Jaume Pujol, and Merc`e Villanueva Classic and Quantum Error Correcting Codes ....................... 56 Santos Gonza´lez, Consuelo Mart´ınez, and Alejandro P. Nicol´as Evaluating the Impact of Information Distortion on Normalized Compression Distance ............................................ 69 Ana Granados, Manuel Cebria´n, David Camacho, and Francisco B. Rodr´ıguez Codes from Expander Graphs ..................................... 80 Tom Høholdt Adaptive Soft-Decision Iterative Decoding Using Edge Local Complementation ................................................ 82 Joakim Grahl Knudsen, Constanza Riera, Matthew G. Parker, and Eirik Rosnes Minimal Trellis Construction for Finite Support Convolutional Ring Codes .......................................................... 95 Margreta Kuijper and Raquel Pinto On the Quasi-cyclicity of the Gray Map Image of a Class of Codes over Galois Rings .................................................... 107 Carlos Alberto Lo´pez-Andrade and Horacio Tapia-Recillas Algebraic Geometry Codes from Castle Curves ...................... 117 Carlos Munuera, Alonso Sepu´lveda, and Fernando Torres VIII Table of Contents Two-PointCodes on Norm-Trace Curves............................ 128 Carlos Munuera, Guilherme C. Tizziotti, and Fernando Torres Close Encounters with Boolean Functions of Three Different Kinds..... 137 Matthew G. Parker Wiretapping Based on Node Corruption over Secure Network Coding: Analysis and Optimization ........................................ 154 Mohammad Ravanbakhsh, Mehdi M. Hassanzadeh, and Dag Haugland On the Kronecker Product Construction of Completely Transitive q-Ary Codes..................................................... 163 Josep Rifa` and Victor Zinoviev Quaternary Unequal Error Protection Codes ........................ 171 Ludo Tolhuizen Communication on Inductively Coupled Channels: Overview and Challenges ...................................................... 186 Øyvind Ytrehus Author Index.................................................. 197 A Diametric Theorem in Zn for Lee and Related m Distances Rudolf Ahlswede1 and Faina I. Solov’eva2 1 Universit¨at Bielefeld, Fakult¨at fu¨r Mathematik, Postfach 100131, 33501 Bielefeld, Germany [email protected] 2 Sobolev Instituteof Mathematics and Novosibirsk State University, pr.ac. Koptyuga4, Novosibirsk 630090, Russia [email protected] Abstract. Wepresentthediametrictheoremforadditiveanticodeswith respecttotheLeedistanceinZn2k,whereZ2k isanadditivecyclicgroup of order 2k. We also investigate optimal anticodes in Zn for the homo- pk geneous distance and in Znm for theKrotov-typedistance. 1 Introduction In this paper we establish the diametric theorem for optimal additive anticodes in Zn2k with respect to the Lee distance, where Z2k is any additive cyclic group of order 2k. We also study additive anticodes for related distances such as the homogeneous distance, see [7], and the Krotov-type distance, see [13]. Farrell [8], see also [15], has introduced the notion of an anticode (n,k,d) as a subspace of GF(2)n with diameter constraint d (the maximum Hamming distance between codewords) and dimension k. In fact earlier anticodes were used by Solomon and Stiffler [16] to construct good linear codes meeting the Griesmer bound, see also [6]. Such anticodes may contain repeated codewords. Like in [1] we study anticodes without multiple codewords. The notion of an optimal anticode investigated in the paper is different from the notion in [15], Chapter17.LetGn be the directproductofn copiesofafinite groupGdefined on the set X ={0,1,...,q−1}. We investigate AGn(d)=max{|U|:U is a subgroup of Gn with D(U)≤d}, whereD(U)= max d(u,u(cid:4))isthediameterofU,d(·,·)istheHammingdistance u,u(cid:2)∈U for any finite group G, the Lee distance or the homogeneous distance for any cyclic group Zpk, where p is prime, or a Krotov-type distance for Znm. In [4] the complete solution of the long standing problem of determining max{|U|:U ⊂Xn with D (U)≤d}, H for the Hamming distance d, is presented and all extremal anticodes are given. AnotherdiametrictheoreminHammingspacesforgroupanticodesisestablished A.Barbero(Ed.):ICMCTA,LNCS5228,pp.1–10,2008. (cid:2)c Springer-VerlagBerlinHeidelberg2008 2 R. Ahlswede and F.I. Solov’eva in[1]:foranyfinitegroupG,everypermittedHammingdistanced,andalln≥d subgroups of Gn with diameter d have maximal cardinality qd. In Section 2 we give necessary definitions and auxiliary results from [1], in Sections3and4weprovethe diametrictheoremforZn withrespecttothe Lee 2k distance,in Section5 we investigate optimalanticodes in Zn endowedwith the pk homogeneous distance, and Section 6 is devoted to optimal anticodes in Zn for m Krotov type distances. 2 Preliminary Definitions and Auxiliary Results Throughout in what follows we consider groups additive and write the concate- nation of words multiplicative, i.e. for un ∈ Zn we use un = u u ...u . The m 1 2 n all-zero word of length n is denoted by 0n. Definition 1. For any U ⊂Xn and S ⊂X, where S (cid:6)=∅, we define US ={u1...un−1 :u1...un−1s∈U for all s from S and u1...un−1s(cid:6)∈U for all s from X (cid:2)S}. From this definition we have the property US ∩US(cid:2) =∅ if S (cid:6)=S(cid:4). (2.1) Definition 2. For any U ⊂Xn we define U(n) ={un ∈X : there exists a word u1...un−1 such that u1...un−1un ∈U}. For two sets U,V ⊂Xn their cross-diameter is defined as D(U,V)= max d(u,v). u∈U,v∈V Let G be any finite Abelian group. Denote by S a subset of G containing 0. 0 Further we will use the following three lemmas, which can be found in [1]. Lemma 1. For any subgroup U of Gn (briefly U < Gn) a non-empty subset U{0}0 of U is its subgroup. Lemma 2.(GeneralizationofLemma1)If U <Gn then for a non-emptysubset US00 from U it is true that US00≤U. Lemma 3. If U is a subgroup of Gn, then (i) There is exactly one subset S0 in G with US0 (cid:6)=∅; (ii) The set S is a group; 0 (iii) The set US0S0 is a subgroup of U. By Lemma 3 we have US0S0 ≤U, so we candecompose a groupU into cosets of the subgroup US0S0: (cid:2) U = (US0 +α)(S0+ψ(α)) (2.2) α for suitable ψ.
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