Applicable Algebra, Error-Correcting Codes, Combinatorics and Computer Algebra: 4th International Conference, AAECC-4 Karlsruhe, FRG, September 23–26, 1986 Proceedings

221 Pages·1988·4.11 MB·English

by  Thomas Beth (auth.)

Lecture Notes ni
Computer Science
Edited yb .G Goos dna .J Hartmanis
307
IIII III
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Springer-Verlag
nilreB grebledieH NewYork nodnoL Paris oykoT
Editorial Board
D. Barstow W. Brauer R Brinch Hansen D. Gries D. Luckham
C. Moler A. Pnueli G. SeegmtJIler .J Steer N. Wirth
srotidE
Beth Thomas
Michael Clausen
Fakult~t riCf Informatik, t.§tisrevinU ehurslraK
Haid-und-Neu-Stra6e 7, D-7500 Karlsruhe ,1 FRG
CR (1987): Subject Classification E.4, .I 1
ISBN 3-540-19200-X Heidelberg Berlin Springer-Verlag New kroY
ISBN 0-387-19200-X galreV-regnirpS New Heidelberg Berlin York
Library of Congress Cataloging-in-Publication Data. AAECC-4 (1986: Karlaruhe, Germany)
Applicable algebra, error-correcting codes, combinatorics and computer algebra. (Lecture notes
ni computer science; 307) Bibliography: p. Includes index. .1 Error-correcting codes (Information
theory)-Congresses. 2. Algebra-Data processing-Congresses. 3. Algorithms-Congresses.
.I Beth, Thomas, 1949-. .1I Clausen, Michael. III. Title. IV. Series.
QA268.A35 1986 005.7'2 88-12246
ISBN 0-387-19200-X (U.S.)
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© Springer-Vertag Berlin Heidelberg 1988
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Preface
The present volume contains the proceedings of the conference AAECC-4 held at Karlsruhe,
September 23 - 26, 1986. AAECC conferences have been organized regularly since 1983:
AAECC-I: Toulouse 1983
AAECC-2: Toulouse 1984
AAECC-3: Grenoble 1985
AAECC-4: Karlsruhe 1986
AAECC-5: Menorca 1987
The next conferences will be held at Roma (1988), Toulouse (1989) and Yokohama (1990).
The proceedings of the previous conferences were published in 1985 [Discr.
Math.56,Oct.1985] and 1986 [Springer LNCS 228, 1986] [Springer LNCS 229, 1986].
Quoting from Jacques Calmet's preface to the AAECC-3 proceedings:
...The main motivation for this series of conferences was to gather researchers in
error-correcting codes, applied algebra and algebraic algorithms. The latter topic has been
extended to computer algebra in general. Applied algebra must be understood as applied to
computer science. After three conferences, it appears that they fitl a communication gap. It is
thus natural that the AAECC conferences areg oing to be held annually in different countries.
For this reason, a permanent organizing committee has been set up. It consists of Thomas
Beth, Jacques Calmet, Anthony C. Hearn, Joos Heintz, Hideki Imai, Heinz Liineburg, H.F.
Mattson Jr. and Alain Poli ....
For AAECC-4, 28 papers (plus 4 invited lectures) were presented during the conference. In
order to guarantee scientific journal standards, in a seconrde viewing processw hich has taken
most of the time since the conference each paper was reviewed a second time by a group of at
least two independent international experts.
We are grateful to the many people who helped us to organize this conference in such a short
length of time. We are especially indebted to Angelika Remmers for her help and support
before, during and after the conference. We are also most grateful to the referees whose
contributions ensure the significance and quality of the papers published in this volume.
Karlsruhe, February 1988 Thomas Beth
Michael Clausen
GENERAL CHAIRMAN:
T. BETH, University of Karlsruhe, FRG
SCIENtIFIC CHAIRMAN:
H. LfJNEBURG, University of Kaiserslautem, FRG
ORGANIZING COMMITTEE:
T. BETH, University of Karlsruhe, FRG
J. CALMET, LIFIA, Grenoble, France
A.C. t-lEARN, The Rand Corp., USA
J. HEINTZ, FRG, Argentina
H. Ig~. I, University of Yokohama, Japan
.H LUNEBURG, University of Kaiserslautern, FRG
H.F. MATTSON jr., Syracuse University, USA
A. POLI, University P.Sabatier, Toulouse, France
SCIENTIFIC COMMITTEE:
P. CAMION, ]ILIA, Le Chesnay, France
G.E. COLLINS, Ohio State University, Colombus, USA
B. COURTEAU, Sherbrooke University, Canada
R. LOOS, Universit~it Karlsruhe and Tiibingen after 1986 1.Oct.
A. MIOLA, University "La Sapienza", Roma, Italy
W. PLESKEN, Queen Mary College, London, UK
H. ZASSENHAUS, Ohio State University, Colombus, USA
LOCAL ORGANIZATION at the University of Karlsruhe:
M. CLAUSEN
A. FORTENBACHER
D. GOLLMANN
A. REMMERS
V. SPERSCHNEIDER
SESSION CHAIRPERSONS:
T. Beth, J. Calmet, J. Heintz, H. Liineburg, H. Imai, R. Loos, A. Poli
LIST OF REFEREES:
T. Beth, B. Buchberger, F. Bundschuh, J. Calmet, P. Camion, M.Clausen, G. Collins,
B. Courteau, A. Diir, J. yon zur Gathen, D. Gollmann, D.Jungnickel, A. Kerber, R.
K6nig, W. Kiichlin, U. Kulisch, J. van Lint, H. Lfineburg, R. Loos, J. Massey, H.F.
Mattson, H. Meyr, A. Miola, H.M. M6Iler, A. Neumaier, H. Niederreiter, D. Perrin, F.
Piper, A. Poli, W. Plesken, A. Sch6nhage, A. Shamir, J.J. Seidet, NJ.A. Sloane, V.
Strassen, W. Trinks
Contents
noitcudortnI
Th.Beth
to Graphs Triangular T(m) From gnitcerroC-rorrE-elgniS Codes
J.M. Basart, L. Huguet suomonotuA( University )anolecraB
Integration of System Algebra Computer a in Tools Graphical 31
.G" Bittencourt (LWIA, )elbonerG
Algebra Computer in Unification Inference Using Type 25
J.Calmet, H. Comon, D.Lugiez (LIFIA, )elbonerG
(Extended Abstract)
Distribution Weight The of gnitcerroC-rorrE-elbuoD Goppa sedoC 29
A. Diir (University of )kcurbsnnI
A Analysis Simple of Algorithm Decoding Blokh-Zyablov the 43
Th. Ericson (University of )gnip6kniL (invited)
Design of with Decoder Viterbi a desaB-rossecorporciM Serial
noitatnemelpmI 58
F.-J. Garcia-Ugalde, R.H. Morelos-Zaragoza A.
ytisrevinU( of )ocixeM
nO s-Sum-Sets (s odd) and Codes Projective Three-Weight 68
M. Griera suomonotuA( University )anolecraB
nO eht Integration of Algebraic and Numeric snoitatupmoC 77
G. Mascari oilgisnoC( delle Nazionale ,ehcrehciR Roma)
A. Miola (University aL" ,"azneipaS Roma)
System Manipulation A Symbolic for Problems Combinatorial 88
P. Massazza, G. Mauri, P. Righi, M. Torelli (University of )onaliM
Bases Standard and Rings Polynomial Non-Commutative Non-Noetherianity: 98
T. Mora (University of )avoneG
Codes Self-Dual Embedding into Space Projective 011
H. Opolka (University of )negnittSG
IV
ehT Finite mrofsnarT-reiruoF Modular a as noitutitsbuS 311
H. Opolka (University of G6ttingen)
Representations Modular and Computers On of ) SLn(K 021
M. Pittaluga, E. Striekland Vergata", Roma) "Tor (University
Construction of GF(2) over SDMC Codes or GF(3) 031
A. Poli Toulouse) P. Sabatier, (University
Program Software A Fast16: for gnisirotcaF over GF(p) Large Polynomials 931
A. Poli, M.C. Gennero Toulouse) P. Sabatier, (University
noitaziretcarahC of yletelpmoC Codes Regular hguorht laimonyloP-P
Schemes Association 751
J. Rifh, L. Huguet Barcelona) University (Autonomous
Curves Hermitian on Codes 861
M.A. Shokrollahi (University of Karlsruhe)
Symmetries of Goppa Extended Cyclic Codes a F over 177
J.A. Thiong-Ly Mirail", "Le (University Toulouse)"
Bounds Some for eht Construction of Bases Gr6bner 591
V. Weispfenning (University of Heidelberg) (invited)
Analysis of a Class of smhtiroglA for Languages Trace on Problems 202
A. Bertoni, M. Goldwurm, N. Sabadini (University of Milano)
Authors Index 2]5
Applicable Algebra, Error-Correcting Codes,
Combinatorics and Computer Algebra
An Introduction to this Volume
Thomas Beth
At a glance the combination of the apparently different fields named in the title seems to be
rather artificial. However, the extensive research in these areas during the last decade has
shown large intersections and interdependencies between these fields, especially towards
several areas of application. These range from coding theory dna signal processing to
software engineering tools based on symbolic manipulation techniques. The use of data
structures with many regularities is the common basis of research in these areas. The
regularities lead to algorithm design principles, from which some of the most eff{cient
computational procedures known in the above areas can be developed.
This volume presents a collection of papers all of which can be viewed as variations on the
intrinsic relation between algorithm engineering and the regularities of the data structures. We
give a short summary of the presented papers. The classification of these papers in one area
only, however, seems to be impractical and infeasible. The subsequent survey and
introduction will show that it is rather appropriate to orientate the contents of this volume in
accordance with the general aims of the AAECC-conference series in form of the following
pentagram:
AE
mhtiroglA gnireenignE
/ \
AA EC
elbacilppA \ arbeglA / gnitcerroc-rorrE sedoC
CA CC
retupmoC Algebra Combinatorics dna
ytixelpmoC
The paper of Bazard and Huguet shows the close relation of single error-correcting codes to
triangular graphs by which a rather regular combinatorial structure is defined in order to
derive with methods of linear algebra the stated results. The connection between regularity of
codes and their presentation by tools of linear algebra is given in the work by Rifa dna teuguH
on the characterization of codes by so-called association schemes. The importance of these
combinatorial structures has been pointed out by Delsarte [9] and has been used since to
determine weight distribution of codes as is also done in the paper by Diir, who determines
the weight distributions of some special Goppa codes. This question is interesting and
important for at least two reasons: on the one hand weight distributions of codes allow for a
rather direct calculation of residual error probabilities in additive symmetric channels. On the
other hand weight distributions can be used to determine possible regularities in the data
structures presented by such codes.
Another example for the close interrelation,between the underlying data structures and the
efficiency of the required algorithms is given in the paper by Ericson on the
Blokh-Zyablov-decoding algorithm. This algorithm has raised quite some interest throughout
the last few years because it seems to be one of the most efficient decoding features applying
the ideas of concatenated codes in a very efficient and clear way. Concatenated codes 1[ t] are
nothing but record-Iike data structures which allow for refined subrecords.
The paper by edlagU-aicraG and azogaraZ-soIeroM yields an implementation of a serial
Viterbi decoding algorithm. The Viterbi decoding algorithm is of high practical significance, it
is an example of dynamic programminogn a data structure which initially is to be considered
as a "hypercube" over GF(2). For further studies, cf. MacWilliams/Sloane 12]. [
The note by Griera relates the algebraic geometric structure of codes to the combinatorial
interpretation via its weight distribution.
Similar in spirit, but quite different in its tools is the paper presented by Opolka on the
embedding of self-dual codes into projective space which demonstrates the close interrelation
between coding theory and representation theory. At this point it should be mentioned how
important the toot of group theory is in dealing with these kinds of regular algorithms, el.
Beth [1]. The same observation applies to the paper by Poli discussing the construction of
self-dual multicirculant codes. The structure in these codes which allow for a large set of
automorphisms needs tools from modular representation theory and polynomial algebra in
several variables.
ihallorkohS uses algebraic geometric tools in his paper dealing with the construction of codes
on algebraic curves. This vastly extends the concept of cyclic codes and Goppa codes as has
been done in the area of coding theory for the last two decades, cf. van Lint [11]. His work
shows the need for more complicated data structures in order to approach the bounds for
so-called good codes as had been pointed out by Tsfasman, Zenk and Vladut some years ago
[2]. Thiong-Ly uses in his paper on the symmetries of conventional Goppa codes classical
polynomial arithmetic methods. The last two mentioned papers set forth some trends in
algebraic coding theory.
The paper by Opolka on the connections between finite Fourier transforms and modular forms
points out another aspect of the importance of classical algebraic tools in dealing with the
questions of coding theory from a higher point of view. The relation to Fourier transforms
shows again the use of methods from group representation theory.
The paper by Pittaluga dna Strickland is concerned with the computation of the (modular)
irreducible K-representations of the special linear groups SL(n,K) over an infinite field K.
The contribution by Righi Massazza, Mauri, dna Torelli presents a symbolic manipulation
system for the solution of counting problems in terms of generating functions. This is an
interesting example of the importanee of computer algebra systemsf or the research in other
fields. Some systems like MACSYMA [3], "Scratchpad" [4] and SAC-2 [5] are provided for
such symbolic manipulations. The tools of software engineering applied here are closely
related to the algebraic combinatorialm ethods by which the problem specification takes place.
Similar aspectsa re dealt with in the papebry Mascari and Miola on integration of numeric and
algebraic computations. Designing such computer algebra systems requires intensive research
in the area of abstract data types and in order to automatize the problem specification a
mechanism for type inference is needed in future systems. The paper by Comon Calmet, and
Lugiez presents a short account of an approach to this problem. The design of scientific
workstations is also related to the availability of graphical tools and their connection to the
underlying computer algebra system. This question is addressed by Bittencourt in his paper
aboutth e integration of graphical toolsi n a computer algebra system.
In this context the paper by Poli dna Gennero shows a very efficient software program for
factorization of polynomials over large finite fields. A reduction relation together with a
Grtbner basis provides a data structure for computing with ideals in polynomial rings.
Weispfenning gives some bounds on the Buchberger algorithm computing a Grtbner basis
dna Mora extends some results on Grtbner bases to the non-commutative case.
Trace languages, closelrye lated to the combinatorial theory of rearrangements, are the subject
of the paper by Goldwurm Bertoni, dna .inidabaS The authors study the time complexity of a
class of algorithmic problems on trace languages both in the worst and in the average case.
This summary has given an overview of the different topics approached in papers in this
volume. The combination of these papers shows one of the main lines of research in the area
of applicable algebra and computer algebra exemplifying the most interesting problem of how
to combine software engineering tools and algorithm engineering tools in order to give
automatical specification and solution methods into the hand of users who are working in
application areas in which regular data structures and efficient algorithms are of large
importance.
References:
]1[ T. Beth: Teubner 1984 Fourier-Transformation, schneUen der Verfahren
]2[ M.A. Tsfasman; Modular Curves, Shimura Curves, and Goppa Codes the better than
S.G. Vladut; T. Zink: Math. Nachr. Varshamov-Gilbert-Bound, 104, 13-28, 1982
]3[ Pavelte, R.; Wang, P.S.: MACSYMA from F to G, J. Symb. Comp. ,1.,)5891( 69-100
]4[ R.D. Jenks; R.S. Sutor; SCRATCHPAD II: An abstract data type system for mathematical
S.M. Watt: computation, in: (ed.): Trends Rdanl~en in Computer Algebra,
International Symposium Bad Neuenahr, Proceedings LNCS 296,
8891
]5[ SAC-2: by G. Collins, Ohio State University; R. Loos, University of
Ttibingen
]6[ AAECC- Proceedings :1 Discrete Mathematics, ,if.5,.loV No. 2-3, Oct. 1985
]7[ Proceedings AAECC-2: Algorithmics Algebra, Applied and Codes, Error-Correcting
(ed. by A. Poli), Springer LNCS 228, 1986
]8[ AAECC-3: Proceedings Algebraic Algorithms and Codes (ed. Error-Correcting by Calmet), J.
Springer LNCS 229, 1986
]9[ P. Delsarte: An Algebraic Approach to Coding Theory, Philips Res.Rep.Supp. ,_0_1
3791
[10] N.J.A. Sloane: Error-Correcting Codes and Invariant Theory, A.M.M, Feb.1977,
82-107
[11] J.H. van Lint: Introduction Coding to Theory, GTM 86, Springer 1982
[12] F.J.MacWilliams;
N.J.A. Sloane: The Theory of Holland North Codes, Error-Correcting 9791