Asymptotic Combinatorial Coding Theory

115 Pages·1997·8.119 MB·English

by  Volodia Blinovsky (auth.)

ASYMPTOTIC COMBINATORIAL
CODING THEORY
THE KLUWER INTERNATIONAL SERIES
IN ENGINEERING AND COMPUTER SCIENCE
ASYMPTOTIC COMBINATORIAL
CODING THEORY
by
Volodia Blinovsky
Institute for Information Transmission Problems,
Russian Academy ofS ciences
SPRINGER SCIENCE+BUSINESS MEDIA, LLC
Library of Congress Cataloging-in-Publication Data
A C.I.P. Catalogue record for this book is available
from the Library of Congress.
ISBN 978-1-4613-7839-6 ISBN 978-1-4615-6193-4 (eBook)
DOI 10.1007/978-1-4615-6193-4
Copyright © 1997 Springer Science+Business Media New York
Originally published by Kluwer Academic Publishers in 1997
Softcover reprint of the hardcover 1s t edition 1997
AU rights reserved. No part of this publication may be reproduced, stored in a
retrieval system or transmitted in any form or by any means, mechanical, photo
copying, recording, or otherwise, without the prior written permission of the
publisher, Springer Science+Business Media, LLC.
Printed on acid-free paper.
Contents
Preface VII
Introduction XI
V%dia B/inovsky
1. CODING BOUNDS 1
2. LIST DECODING 7
3. COVERING AND PACKING 41
4. DECODING COMPLEXITY 63
5. CHANNEL WITH DEFECTS 75
6. SOME OTHER PROBLEMS 79
Index
89
References 91
v
Preface
The coding theory originated in forties. Pioneer worksofC.Shannon contain
inition notions of the information theory and the asymptotic combinatorial
coding theory as its part. However, at present the combinatorial coding
theory is an independent area of investigations with its own methods. The
object of investigation in the coding theory is the code. A code is the finite
set in a designated space. The main problem in the coding theory was
the determination of the cardinality of the code in metric space with given
minimum distance over pairs of points from the code. Nowdays, asymptotical
combinatorialcodingtheory dealswithallnotionsand methodsofcombinatorics
and provides an undestanding ofinvestigations ofdiscrete probabilistic spaces,
covering problems, the graph theory, the source coding, the complexity theory,
and many other problems. It is necessary to note that many problems in the
combinatorialcoding theory are difficult to solve and their solution is the result
oflongstandinginvestigations. The 'main'problemin the combinatorialcoding
theory is to prove the tightness of the asymptotic Varshamov-Gilbert bound
for a binary code and this problem is still open.
This book collect some problems of the asymptotical combinatorial coding
theory. 'Asymptotical' means that we are interested in the behavior of some
functions when their parameters (usualy 'length' ofthe code and some others)
tend to infinity. Here probabilistic and analytic methods are used in addition
to combinatorial methods. Our aim was to introduce the most interesting
problems which in some sense are solved completely. Here we collect such
problems almost all of which are investigated or solved by the scientists from
Russia and where not published in monographs before (except the two first
introductory chapters).
VII
Vlll ASYMPTOTIC COMBINATORIAL CODING THEORY
In Introduction and Chapt.l we describe a number ofknown facts and prove
those ofthem which we use in the subsequent chapters. Chapter 2contains the
coding theorem and the bounds ofthe reliabilityfunction in list decoding case.
In Chapt.3 we introduce problems concerning the estimation of the fraction
of codes, which attain the Goblick bound. We also prove there the existence
of asymptotically optimal coverings and packings of certain Hamming spaces
(with the coordinates of vectors from a finite field) by balls of unit radius.
Chapter 4 contains the combinatorial algorithm of suboptimal decoding and
half distance decoding of the codes attaining the Varshamov-Gilbert bound.
Chapter 5 gives the construction ofthe lower bound ofthe cardinality ofcode,
which correct the defects and errors. Chapter 6 contains the solutions of two
problems: to find the capacity of pentagon and to derive the formula for the
minimal price ofthe code on the combinatorialsources.
We hope that the present book will be of interest for a wide range of
investigators in combinatorics and students interested in the combinatorial
coding theory. The material of the book can be used in a special course of
combinatorics.
To undestand the present text it is not necessary to resort to other soucers
since all necessary auxiliary results and their proofs are given in Introduction.
The material of all chapters except Chapts. 4 and 5 can be understood using
the facts from Introduction. To understand Chapts.4 and 5 it is necessary to
be acquainted with the material of Introduction and Chapter 3.
At the end of the chapters we give some problems. Unsolved problems are
marked by the asterisk.
The book benefited by numerous discussions with colleagues, most notably
L. Bassalygo, R. Dobrushin and M. Pinsker.
VOLODIA BLINOVSKY
IX
Notes
1. This work supported by International Science Foundation Grant no.M5E300
INTRODUCTION
Volodia Blinovsky
All sets considered here are finite. For an arbitrary set F consisting of q
q
elements, we denote by F; the space ofall sequences with elements from F of
q
length n with Hamming distance d(.,.) which is defined by the equality
n
2:(1-
d(x, y) ~ bXi,y,);
i=l
Thespace F; iscalled the Hammingspace. By Ix 1A= d(O-,x);0- = (0, ...,0),x E
F;,
we mean the Hamming weight of x. In most cases we suppose that the
set F is the field of q elements and use the same notation as above. In the
q
=
case when q 2 we omit q in the notation. The distance between two sets
A,B C F; is defined by the equality
=
d(A, B) min d(x, y).
xEX;yEY
Denote C~ ~ n(n-l1.)n-i+l1, where n,i E {O, 1,...} (it is called the binomial
coefficient). Let rZ1(lzJ), z E R1, be the maximal (minimal) integer which
~
does not exceed (is not less than) z. Denote by ( ) the set of k-element
subsets of the set A. There exists the natural bijection between Fn and
2un =A Unk=O (Ukn) ' where Un =A {1,2, ...,n}. Let x = (Xl,""Xn) E
n
F ;Xii' ...,Xik = 1, and let other coordinates of x be zero. Let Tx =
{ii,...,ik}, then the bijection is determined by the relation x ..... Tx. Then
d(x,y) =1 TxD.Ty I , where TxD.Ty = (Tx \ Ty)U(Ty \ Tx); x,y E Fn. Let
Wnq(r) = C~(q-It be the volumeofthe sphere Cnq(z,r) =A 2:xEF.;';d(x,zl=r x
Xl
Xli ASYMPTOTIC COMBINATORIAL CODING THEORY
with center in z E F; and let Vnq(r) = 2:~=0 C~(q - l)t be the volume ofthe
ball Bnq(z, r) ~ 2:~=0Cnq(z, i) with center in z E F;. We use the notation
o(n) ,lo(n)1 --+ oo,o(n)jn --+ 0,n --+ 00, which can be different in different
formulas. We set Cl'(n) ::::: (3(n) iff 1Cl'(n) 1jC ~I (3(n) I~ C 1Cl'(n) 1for some C
and large enough n; Cl'(n) =O((3(n)) iff 1Cl'(n) I~ C 1(3(n) Iand large enough n;
=
Cl'(n) '" (3(n) iff limn_ ooCl'(n)j(3(n) 1. Here Cl'(n),(3(n) are some sequences.
Consider the following problem: let 5 c Fn be the set with cardinality
r
=LC~.
151= Vn(r)
i=O
Consider a new set
(L- neighborhood ofthe set 5).
Statement 1 (Harper, Katona) The following relation is valid:
(1.1 )
The proofofthis statement can befound, for example, in [Bolobas, 1986] and it
is introduced here for completeness. In the case when r +L ~ n the statement
is evident. Suppose that r + L < n. Let 51 be the set on which the min
in (1.1 ) is achieved and 52 = Fn\DL(5I). The set 52 consists of points
s E Fn whose distance from 51 exceeds L. Let us show that it is possible°to
n
choose 51 = Bn(1, r),1= (1, ...,1) E F ,then 52 = Bn(O,n - r - L -1), =
n
(0, ...,0) E F . Suppose that 51:/; Bn(1,r). Let
Lis
p(5 52) = Lis 1- I·
1,
sES, SES2
Note. It iseasy toshow that ifs E Fn,then for 51 ={sU SI;SI E 51, SU SI ~
5dU{SI; SI E 51,sUSI E 5d and52 ={S2\S;S2 E 52 S2\S ~ 52}U{s2; S2 E
52, S2 \ SE 52} the relations
are valid.
Let us show that there exist sets 5~,5~ such that d(5L 5~) > L, 15~ =
1
1511,15~1 =1521and p(51,52) < p(5~,5~). Since the number p(51,52) is an