Table Of ContentLecture Notes in Computer Science 5228
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UniversityofCalifornia,LosAngeles,CA,USA
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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]
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ISSN 0302-9743
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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.
