Perfect Codes And Related Structures

436 Pages·2022·7.085 MB·English

by  Tuvi Etzion

PERFECT CODES
AND RELATED STRUCTURES
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PERFECT CODES
AND RELATED STRUCTURES
Tuvi Etzion
Technion, Israel
NEW JERSEY • LONDON • SINGAPORE • BEIJING • SHANGHAI • HONG KONG • TAIPEI • CHENNAI • TOKYO
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PERFECT CODES AND RELATED STRUCTURES
Copyright © 2022 by World Scientific Publishing Co. Pte. Ltd.
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To
Orna, Ophir, Liron, and Reut
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Preface
Information theory, launched by the pioneering work of Shannon in 1948,
has generated a lot of applications and ten of thousands of research pa-
pers. Onecaneasilysay, withoutstretchingthetruth, thatitsinfluenceon
our daily lives is pervasive. Coding theory, which is one of the important
sub-areasofinformationtheory, startedwiththeworkofGolay(1949)and
Hamming (1950). This research area was motivated by engineering prob-
lems,andfrom1950untiltoday,withthegrowthofdigitalcommunication,
thedemandforoldandnewtechniquesincodingtheoryhasonlyincreased.
Although some basics of the theory are not very difficult, over time more
andmoresophisticatedmathematicshasbeenusedanddevelopedincoding
theory. Thishasmadetheareaofcodingtheoryveryimportanttoelectrical
engineers and to computer scientists on one hand and to mathematicians
on the other hand.
Two of the most important types of codes are error-correcting codes
and covering codes. In the first thirty years of this research area, most
work done in coding theory was related to error-correcting codes. Several
textbooks and monographs were written in this research area. It was no
surprise that also undergraduate and graduate courses, in this area, were
developed in many universities. Forty years ago, as well, the area of cov-
ering codes started to develop, with hundreds of research papers and new
applications that were found quite frequently. Perfect codes, which are the
codes considered in our book, lie exactly in the intersection between error-
correctingcodesandcoveringcodes. Theywereconsideredinallthebooks
on coding theory, but were never the highlight of these books as they are
in our book.
TheexistingbooksoncodingtheoryfocusmainlyoncodesintheHam-
ming space. In some books, there are one or two chapters related to other
vii
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viii Perfect Codes and Related Structures
metrics, such as codes in the Lee metric or constant-weight codes in the
Johnson metric. Nevertheless, the codes considered for other metrics were
usually not perfect. Moreover, in these books there are one or at most
two chapters devoted to perfect codes and related structures. Our book is
different. First, it is devoted solely to perfect codes of various types. It is
alsodifferentaswedevoteafewchapterstotheHammingspace, butother
chapters, whichcomprisethegreaterpartofthebook, aredevotedtocom-
binatorial designs, mixed codes, constant-weight codes, to the Lee metric,
to the Grassmann scheme, to metrics related to storage devices, as well as
to other metric spaces and codes related to newly important applications.
Moreover, also in the Hamming space, we consider a few topics that are
not covered in other books.
We tried to make the book as self-contained as possible, providing de-
tailed proofs, many times more detailed than the ones that appear in the
literature. The detailed proofs will enable to use this book also as a text-
book for courses in coding theory. In many cases, no proofs for known
results were provided in the cited papers and the appropriate proofs are
given in our book. In some cases, we present results for the first time. Of
course, we could not provide details on every important technique known
in the literature and, in some cases, we offer only pointers to reference
material. These references and a summary of these techniques are usually
provided in the notes that form the last section of each chapter. Notably,
therearemanytechniquestoprovethenonexistenceofperfectcodesinthe
various metrics. Therefore, we concentrate on comprehensive treatment of
such techniques for one metric only, the Johnson metric.
Our intension is to offer a different perspective for the area of perfect
codes. For example, in many chapters there is a section devoted to diam-
eter perfect codes. In these codes, anticodes are used instead of balls and
these anticodes are related to intersecting families, an area that is part of
extremal combinatorics. This is one example that shows how we direct
our exposition in this book to both researchers in coding theory and math-
ematicians interested in combinatorics and extremal combinatorics. New
perspectives for MDS codes, different from the classic ones, which lead to
new directions of research on these codes are another example of how this
bookmayappealtobothresearchersincodingtheoryandmathematicians.
Our point of view is mainly combinatorial and hence some of the algebraic
approach will be omitted (also for lack of space).
The book is so that it can be used by a beginner who just wants either
to learn something about prefect codes or to conduct research on perfect
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Preface ix
codes. Nevertheless, it can be also used by the more advanced reader who
hassomeknowledgeoncodingtheoryandwantstogetsomeinformationon
perfectcodesortofindsomenewlinesofresearchinthisarea. Throughout
thebook,therearemanyresearchproblems,someofwhichwethinkcanbe
usedtomotivategraduatestudentsandsomewhichareextremelydifficult.
We would be very happy to see the book’s reader make a breakthrough as
a result of insights gleaned through reading it. This book is a monograph;
as we did not introduce exercises and assignments for the reader. It was,
however, written in a way that enables it to also be used as a textbook
for either a basic combinatorial course in coding theory for undergraduate
students in mathematics, as an advanced course for the same students, or
for an advanced course in coding theory, for students in computer science
orelectricalengineering,whichemphasizesperfectcodes. Eachsuchcourse
should be based on different chapters of this book. This is the main reason
that we have provided proofs for most of the lemmas and theorems in the
book. Moreover, references are quoted only in the notes of each chapter
and not along the various sections of the chapter, although some isolated
proofs use results that are provided either without proofs or in the notes
with or without proofs.
Asabasiccourseforundergraduatestudentsinmathematics,wesuggest
using the first seven chapters (excluding Chapter 1), which are devoted
to the Hamming metric and to combinatorial designs, and Chapter 11,
which focus on the Lee metric. The other chapters can be used in an
advancedcourseformathstudents. Asforanadvancedcourseforstudents
in computer science and electrical engineers, who have already completed
a basic course in coding theory, we suggest taking highlights from each
chapter,withthepossibleexceptionofthefirstchapter. Inouropinionmost
chapters cannot be taught in only two hours. If the course compromises
on thirteen weeks, we suggest splitting Chapter 3 into two weeks. We also
suggest combining Chapter 4 and Chapter 5 and cover them in two classes
since Chapter 4 is quite light, while Chapter 5 has lot of material.
A large part of this book is based on my own research work, which was
performed over the last thirty years. My Ph.D. advisor, Abraham Lempel,
guided me as I took my first steps into coding theory. My post-doc advi-
sor,SolomonW.Golomb,introducedmetocombinatorialdesignsandtheir
intersectionwithcodingtheoryingeneralandwithperfectcodesinparticu-
lar. IwasfirstintroducedtosomeperfectcodesproblemsbyGerardCohen
and Simon Litsyn. The discussions I had with them led to my first paper
on perfect codes with Alexander Vardy. This was our first joint paper and