Graduate Texts in Mathematics 86

Graduate Texts in Mathematics 86

Springer-Verlag Berlin Heidelberg GmbH

Editorial Board S.Axler F. W. Gehring K.A.Ribet

T AKEUTI/ZARING. Introduction to Axiomatic 33 HIRSCH. Differential Topology.

Set Theory. 2nd ed. 34 SPlr.t.ER. Principles of Random Walk. 2nd ed.

2 OXTOBY. Measure and Category. 2nd ed. 35 WERMER. Banach Algebra.~ and Several 3 SCHAEFFER. Topological Vector Spaces. Complex Variables. 2nd ed.

4 HILTON/STAMMBACH. A Course in 36 KELLEy/NAMIOKA et al. Linear Homological Algebra. 2nd ed. Topological Spaces.

5 MAC LANE. Categories for the Working 37 MONK. Mathematical Logic.

Mathematician. 2nd ed. 38 GRAUERT/FRIT.lSCHE. Several Complex 6 HUGHES/PIPER. Projective Planes. Variables.

7 SERRE. A Course in Arithmetic. 39 ARVESON. An Invitation to C*-Algebra.~.

8 TAKEUTl/ZARING. Axiomatic Set Theory. 40 KEMENy/SNEUJKNAPP. Denumerable Markov 9 HUMPHREYS. Introduction to Lie Aigebras and Chains. 2nd ed.

Representation Theory. 41 ApOSTOL. Modular Functions and Dirichlet 10 COHEN. A Course in Simple Homotopy Series in Number Theory. 2nd ed.

Theory. 42 SERRE. Linear Representations of Finite

II CONWAY. Functions ofOne Complex Groups.

Variable I. 2nd ed. 43 GILLMAN/JERISON. Rings of Continuous 12 BEALS. Advanced Mathematical Analysis. Functions.

13 ANDERSON/FULLER. Rings and Categories of 44 KENDlG. Elementary Algebraic

Modules. 2nd ed. Geometry.

14 GOLUBITSKy/GUILLEMIN. Stable Mappings and 45 LoilvE. Probability Theory 1. 4th ed.

Their Singularities. 46 LoilvE. Probability Theory II. 4th ed.

15 BERBERIAN. Lectures in Functional Analysis 47 MOISE. Geometric Topology in Dimensions

and Operator Theory. 2 and 3.

16 WINTER. The Structure of Fields. 48 SACHs/WU. General Relativity for 17 ROSENBLATI. Random Processes. 2nd ed. Mathematicians.

18 HALMOS. Measure Theory. 49 GRUENBERG/WEIR. Linear Geometry. 2nd ed.

19 HALMOS. A Hilbert Space Problem Book. 50 EDWARDS. Fermat's Last Theorem.

2nd ed., revised. 51 KUNGENBERG. A Course in Differential 20 HUSEMULLER. Fibre Bundles. 2nd ed. Geometry.

21 HUMPHREYS. Linear Aigebraic Groups. 52 HARTSHORNE. Aigebraic Geometry.

22 BARNES/MACK. An Aigebraic Introduction to 53 MANIN. A Course in Mathematical Logic.

Mathematical Logic. 54 GRA VEIlIW ATKINS. Combinatorics with 23 GREUB. Linear Algebra. 4th ed. Emphasis on the Theory of Graphs.

24 HOLMES. Geometric Functional Analysis and 55 BROWN/PEARCY. Introduction to Operator its Applications. Theory 1: Elements of Functional Analysis.

25 HEWITI/STROMBERG. Real and Abstract 56 MASSEY. Aigebraic Topology:

Analysis. An lntroduction

26 MANF$. Aigebraic Theories. 57 CROWEUJFOX. lntroduction to Knot Theory.

27 KELLEY. General Topology. 58 KOBLlTZ. p-adic Numbers, p-adic Analysis, 28 ZARISKI/SAMUEL. Commutative Algebra. and Zeta-Functions. 2nd ed.

VoI. I. 59 LANG. Cyclotomic Fields.

29 ZARISKI/SAMUEL. Commutative Algebra. 60 ARNOLD. Mathematical Methods in Classical

VoI. II. Mechanics. 2nd ed.

30 JACOBSON. Lectures in Abstract Algebra I. 61 WHITEHEAD. Elements of Homotopy Theory.

Basic Concepts. 62 KARGAPOLOV/MERZIJAKOV. Fundamentals of

31 JACOBSON. Lectures in Abstract Algebra II. the Theory of Groups.

Linear Algebra. 63 BOLLOBAS. Graph Theory.

32 JACOBSON. Lectures in Abstract Algebra III. 64 EDWARDS. Fourier Series. VoI. 1. 2nd ed.

Theory of Fields and Galois Theory. continued llfter index

J. H. van Lint

Introduc-tion to Coding Theory

Third Revised and Expanded Edition

, Springer

Eindhoven University of Technology Department of Mathematics

Den Dolech 2, P.O. Box 513 5600 MB Eindhoven The Netherlands

Editorial Board S. Axler

Mathematics Department San Francisco

State University

San Francisco, CA 94132 USA

F. W. Gehring

Mathematics Department University of Michigan Ann Arbor, MI 48109 USA

Library of Congress Catalogmg·in-Publication Data Lint, Jacobus Hendricus van, 1932-

Introduction to coding theory / J.H. van Lin!. -- 3rd rev. and expanded ed.

p. cm. -- (Graduate texts ,n mathematics, 0072-5285 ; 86) Includes bibliographical references and index.

ISBN 978-3-642-63653-0 ISBN 978-3-642-58575-3 (eBook) DOI 10.1007/978-3-642-58575-3

1. Coding theory. 1. Titie. II. Series.

QA26B .L57 1998 003'.54--dc21

98-48080 CIP

K. A. Ribet

Mathematics Department University of California at Berkeley

Berkeley, CA 94720-3840 USA

Mathematics Subject Classification (1991): 94-01, 94B, l1T71

ISSN 0072-5285

ISBN 978-3-642-63653-0

This work is subject to copyright. AII rights are reserved, wether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of iIIustrations, recitation, broadcasting, reproduction on microfilms or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer-Verlag. Violations are liable for prosecution under the German Copyright Law.

© Springer-Verlag Berlin Heidelberg 1982, 1992, 1999

Typesetting: Asco Trade Typesetting Ltd., Hong Kong

46/3111- 5432 - Printed on acid-free paper SPIN 11358084

Preface to the Third Edition

It is gratifying that this textbook is still sufficiently popular to warrant a third edition. I have used the opportunity to improve and enlarge the book.

When the second edition was prepared, only two pages on algebraic geometry codes were added. These have now been removed and replaced by a relatively long chapter on this subject. Although it is still only an introduction, the chapter requires more mathematical background of the reader than the remainder of this book.

One of the very interesting recent developments concerns binary codes defined by using codes over the alphabet 7l.4• There is so much interest in this area that a chapter on the essentials was added. Knowledge of this chapter will allow the reader to study recent literature on 7l.4-codes.

Furthermore, some material has been added that appeared in my Springer Lec- ture Notes 201, but was not included in earlier editions of this book, e. g. Generalized Reed-Solomon Codes and Generalized Reed-Muller Codes. In Chapter 2, a section on "Coding Gain" ( the engineer's justification for using error-correcting codes) was added.

For the author, preparing this third edition was a most welcome return to mathematics after seven years of administration. For valuable discussions on the new material, I thank C.P.l.M.Baggen, I. M.Duursma, H.D.L.Hollmann, H. C. A. van Tilborg, and R. M. Wilson. A special word of thanks to R. A. Pellikaan for his assistance with Chapter 10.

Eindhoven November 1998

I.H. VAN LINT

The first edition of this book was conceived in 1981 as an alternative to outdated, oversized, or overly specialized textbooks in this area of discrete mathematics-a field that is still growing in importance as the need for mathematicians and computer scientists in industry continues to grow.

The body of the book consists of two parts: a rigorous, mathematically oriented first course in coding theory followed by introductions to special topics. The second edition has been largely expanded and revised. The main editions in the second edition are:

(1) a long section on the binary Golay code;

(2) a section on Kerdock codes;

(3) a treatment of the Van Lint-Wilson bound for the minimum distance of cyclic codes;

(4) a section on binary cyclic codes of even length;

(5) an introduction to algebraic geometry codes.

Eindhoven

November ] 99 ] J .H. V AN LINT

Preface to the First Edition

Coding theory is still a young subject. One can safely say that it was born in 1948. It is not surprising that it has not yet become a fixed topic in the curriculum of most universities. On the other hand, it is obvious that discrete mathematics is rapidly growing in importance. The growing need for mathe- maticians and computer scientists in industry will lead to an increase in courses offered in the area of discrete mathematics. One of the most suitable and fascinating is, indeed, coding theory. So, it is not surprising that one more book on this subject now appears. However, a little more justification and a little more history of the book are necessary. At a meeting on coding theory in 1979 it was remarked that there was no book available that could be used for an introductory course on coding theory (mainly for mathematicians but also for students in engineering or computer science). The best known text- books were either too old, too big, too technical, too much for specialists, etc.

The final remark was that my Springer Lecture Notes (#201) were slightly obsolete and out of print. Without realizing what I was getting into I announced that the statement was not true and proved this by showing several participants the book Inleiding in de Coderingstheorie, a little book based on the syllabus of a course given at the Mathematical Centre in Amsterdam in 1975 (M.C. Syllabus 31). The course, which was a great success, was given by M.R. Best, A.E. Brouwer, P. van Emde Boas, T.M.V. Janssen, H.W. Lenstra Jr., A. Schrijver, H.C.A. van Tilborg and myself. Since then the book has been used for a number of years at the Technological Universities of Delft and Eindhoven.

The comments above explain why it seemed reasonable (to me) to translate the Dutch book into English. In the name of Springer-Verlag I thank the Mathematical Centre in Amsterdam for permission to do so. Of course it turned out to be more than a translation. Much was rewritten or expanded,

problems were changed and solutions were added, and a new chapter and several new proofs were included. Nevertheless the M.e. Syllabus (and the Springer Lecture Notes 201) are the basis ofthis book.

The book consists of three parts. Chapter 1 contains the prerequisite mathematical knowledge. It is written in the style of a memory-refresher. The reader who discovers topics that he does not know will get some idea about them but it is recommended that he also looks at standard textbooks on those topics. Chapters 2 to 6 provide an introductory course in coding theory.

Finally, Chapters 7 to 11 are introductions to special topics and can be used as supplementary reading or as a preparation for studying the literature.

Despite the youth of the subject, which is demonstrated by the fact that the papers mentioned in the references have 1974 as the average publication year, I have not considered it necessary to give credit to every author of the theorems, lemmas, etc. Some have simply become standard knowledge.

It seems appropriate to mention a number of textbooks that I use regularly and that I would like to recommend to the student who would like to learn more than this introduction can offer. First of all F.J. MacWilliams and N.J.A. Sloane, The Theory of Error-Correcting Codes (reference [46]), which contains a much more extensive treatment of most of what is in this book and has 1500 references! For the more technically oriented student with an interest in decoding, complexity questions, etc. E.R. Berlekamp's Algebraic Coding Theory (reference [2]) is a must. For a very well-written mixture of information theory and coding theory I recommend: R.J. McEliece, The Theory of Information and Coding (reference [51]). In the present book very little attention is paid to the relation between coding theory and combina- torial mathematics. For this the reader should consult P.J. Cameron and J.H. van Lint, Designs, Graphs, Codes and their Links (reference [11]).

I sincerely hope that the time spent writing this book (instead of doing research) will be considered well invested.

Eindhoven July 1981

J.H. VAN LINT

Second edition comments: Apparently the hope expressed in the final line of the preface of the first edition came true: a second edition has become neces- sary. Several misprints have been corrected and also some errors. In a few places some extra material has been added.

Contents

Preface to the Third Edition . Preface to the Second Edition Preface to the First Edition

CHAPTER 1

Mathematical Background

v

VD IX

1.1. Algebra... 1

1.2. Krawtchouk Polynomials. . . .. 14

1.3. Combinatorial Theory . . . 17

1.4. Probability Theory . . . .. 19

CHAPTER 2 Shannon's Theorem 22 2.1. Introduction... 22

2.2. Shannon's Theorem . . . 27

2.3. On Coding Gain •.••...•..••....•••...••.•••••• 29

2.4. Comments . . . 31

2.5. Problems . . . .. 32

CHAPTER 3 Linear Codes 33 3.1. Block Codes . . . 33

3.2. Linear Codes . . . .. 35

3.3. Hamming Codes . . . .. 38

3.4. Majority Logic Decoding. . . .. 39

3.5. Weight Enumerators . . . .. 40

3.6. The Lee Metric . . . .. 42

3.7. Comments . . . 44

3.8. Problems . . . .. 45

CHAPTER 4 Some Good Codes 47 4.1. Hadamard Codes and Generalizations . . . .. 47

4.2. The Binary Golay Code . . . 48

4.3. The Ternary Golay Code . . . .. 51

4.4. Constructing Codes from Other Codes . . . .. 51

4.5. Reed-Muller Codes . . . 54

4.6. Kerdock Codes . . . .. 60

4.7. Comments . . . 61

4.8. Problems . . . .. 62

CHAPTER 5 Bounds on Codes . . 64

5.1. Introduction: The Gilbert Bound. . . .. 64

5.2. Upper Bounds . . . • . . . 67

5.3. The Linear Programming Bound . . . .. 74

5.4. Comments . . . 78

5.5. Problems . . . .. 79

CHAPTER 6 Cyclic Codes . . . " 81 6.1. Definitions . . . .. 81

6.2. Generator Matrix and Check Polynomial . . . 83

6.3. Zeros of a Cyclic Code . . . .. 84

6.4. The Idempotent of a Cyclic Code . . . 86

6.5. Other Representations of Cyclic Codes . . . 89

6.6. BCH Codes . . . .. 91

6.7. Decoding BCH Codes .. . . . .. 98

6.8. Reed-Solomon Codes . . . . . . .. 99

6.9. Quadratic Residue Codes. . . .. 103

6.10. Binary Cyclic Codes of Length 2n(n odd) . . . .. 106

6.11. Generalized Reed-Muller Codes . . . .. 108

6.12. Comments . . . . 6.13. Problems . . . . CHAPTER. 7 110 111 Perfect Codes and Uniformly Packed Codes . . . 112

7.1. Lloyd's Theorem . . . ; . . . 112

7.2. The Characteristic Polynomial of a Code . . . 115

Contents

7.3. Unifonnly Packed Codes . . . . 7.4. Examples of Unifonnly Packed Codes . 7.5. Nonexistence Theorems

7.6. Comments

7.7. Problems . . . .

CHAPTER 8 Codes over Z4

8.1. Quaternary Codes. . . . 8.2. Binary Codes Derived from Codes over Z4 8.3. Galois Rings over Z4 .

8.4. Cyclic Codes over Z4 . 8.5. Problems . . . .

CHAPTER 9 GoppaCodes

9.1. Motivation . . . . . . . . 9.2. Goppa Codes . . .. . . . . . 9.3. The Minimum Distance of Goppa Codes 9.4. Asymptotic Behaviour of Goppa Codes 9.5. Decoding Goppa Codes

9.6. Generalized BCH Codes 9.7. Comments

9.8. Problems . . . .

CHAPTER 10

Algebraic Geometry Codes 10.1. Introduction. . . 10.2. Algebraic Curves . . . . 10.3. Divisors . . . . 10.4. Differentials on a Curve 10.5. The Riemann-Roch Theorem 10.6. Codes from Algebraic Curves 10.7. Some Geometric Codes . . . .

10.8. Improvement of the Gilbert-Varshamov Bound 10.9. Comments

1O.1O.Problems . . . .

CHAPTER 11

Asymptotically Good Algebraic Codes 11.1. A Simple Nonconstructive Example 11.2. Justesen Codes

11.3. Comments 11.4. Problems . . .

XIII

118 120 123 127 127

128 128 129 132 136 138

139 139 14() 142 143 144 145 146 147

148 148 149 155 156 158 160 162 165 165 166

167 167 168 172 172

CHAPTER 12

Arithmetic Codes 173

12.1. AN Codes. . . .. 173

12.2. The Arithmetic and Modular Weight . . . .. 175

12.3. Mandelbaum-Barrows Codes . • • . . . . • . . • . . . 179

12.4. Comments . . . 180

12.5. Problems . . . • .. 180

CHAPTER 13 Convolutional Codes 181 13.1. Introduction. . • • . . . • . . . • . . . .. 181

13.2. Decoding of Convolutional Codes . • . . . • . . . .. 185

13.3. An Analog of the Gilbert Bound for Some Convolutional Codes . • . • • • •• 187 13.4. Construction of Convolutional Codes from Cyclic Block Codes . . . 188

13.5. Automorphisms of Convolutional Codes. • • • . . . .. 191

13.6. Comments . . . 193

13.7. Problems . . . .. 194

Hints and Solutions to Problems 195 References 218 Index. . . 223

CHAPTER 1

Mathematical Background

In order to be able to read this book a fairly thorough mathematical back- ground is necessary. In different chapters many different areas of mathematics playa role. The most important one is certainly algebra but the reader must also know some facts from elementary number theory, probability theory and a number of concepts from combinatorial theory such as designs and geo- metries. In the following sections we shall give a brief survey of the prerequi- site knowledge. Usually proofs will be omitted. For these we refer to standard textbooks. In some of the chapters we need a large number off acts concerning a not too well-known class of orthogonal polynomials, called Krawtchouk polynomials. These properties are treated in Section 1.2. The notations that we use are fairly standard. We mention a few that may not be generally known. If C is a finite set we denote the number of elements of C by

I CI.

If the expression B is the definition of concept A then we write A := B. We use "iff"

for "if and only if". An identity matrix is denoted by I and the matrix with all entries equal to one is J. Similarly we abbreviate the vector with all coordinates 0 (resp. 1) by 0 (resp. 1). Instead of using [x] we write LXJ :=

max{n E illn ^S;

x}

and we use the symbol

rxl

for rounding upwards.

§ 1.1. Algebra

We need only very little from elementary number theory. We assume known that in '" every number can be written in exactly one way as a product of prime numbers (if we ignore the order of the factors). If a divides b, then we write alb. If p is a prime number and p'la but p,+1

i

a, then we write p'lI a. If J. H. Lint, Introduction to Coding Theory

© Springer-Verlag Berlin Heidelberg 1999

kEN, k > 1, then a representation of n in the base k is a representation

I

n = " ^f..., n·k^{, i ,}

i=O

o

^{~ ni < k for 0 ~ i ~} I. The largest integer n such that nla and nib is called the greatest common divisor of a and b and denoted by g.c.d.(a, b) or simply (a, b). If ml(a - b) we write a == b (mod m).

(1.1.1) Theorem. If

qJ(n) :=

I

{m ^E N

11

~ m ~ n, (m, n) = I}

I,

then

(i) qJ(n) = n TIpln (1 - lip), (ii) Ldln qJ(d) = n.

The function qJ is called the Euler indicator.

(1.1.2) Theorem. If (a, m) = t then a'P(m) == 1 (mod m).

Theorem 1.1.2 is called the Euler- Fermat theorem.

(1.1.3) Definition. The Mobius junction JL is defined by

J1.(n) := ( ^{ -1)\ if n is the product of k distinct prime factors, t, ifn = 1,

0, otherwise.

(1.1.4) Theorem. Iff and g are functions defined on N such that g(n)

= L

^f(d),

din

then

Theorem 1.1.4 is known as the Mobius inversion formula.

Algebraic Structures

We assume that the reader is familiar with the basic ideas and theorems of linear algebra although we do refresh his memory below. We shall first give a sequence of definitions of algebraic structures with which the reader must be familiar in order to appreciate algebraic coding theory.

§ 1.1. Algebra 3 (1.1.5) Definition. A group (G, ) is a set G on which a product operation has been defined satisfying

(i) 'v'aeG'v'beG[ab E G],

(ii) 'v'aeG'v'beG'v'ceG[(ab)c = a(bc)], (iii) 3eeG'v'aeGEae = ea = a],

(the element e is unique), (iv) 'v'aeG3beG[ab = ba = e],

(b is called the inverse of a and also denoted by a-I).

If furthermore (v) 'v'aeG'v'beG[ab = ba],

then the group is called abelian or commutative.

If(G, ) is a group and H c G such that (H, ) is also a group, then (H, ) is called a subgroup of(G, ). Usually we write G instead of(G, ). The number of elements of a finite group is called the order of the group. If (G, ) is a group and a ^E G, then the smallest positive integer n such that an = e (if such an n exists) is called the order of a. In this case the elements e, a, a^{2 , ••• ,} a^{n -1} form a so-called cyclic subgroup with a as generator. U(G, ) is abelian and (H, ) is a subgroup then the sets aH:= {ahJh E H} are called easets of H. Since two cosets are obviously disjoint or identical, the cosets form a partition of G. An element chosen from a coset is called a representative of the coset. It is not difficult to show that the co sets again form a group if we define mUltiplication of co sets by (aH)(bH):= abH. This group is called the Jactor group and indicated by GIR. As a consequence note that if a E G, then the order of a divides the order of G (also if G is not abelian).

A fundamental theorem of group theory states that a finite abelian group is a direct sum of cyclic groups.

(1.1.6) Definition. A set R with two operations, usually called addition and multiplication, denoted by (R,

+, ),

is called a ring if

(i) (R, +) is an abelian group, (ii) 'v'aeR'v'beR'v'ceR[(ab)c = a(bc)],

(iii) 'v'aeR'v'beR'v'ceR[a(b

+

c) = ab

+

ac ^1\ (a

+

b)c = ac

+

bc].

The identity element of (R, +) is usually denoted by O.

If the additional property (iv) 'v'aeR'v'beR[ab

=

ba]

holds, then the ring is called commutative.

The integers 71. are the best known example of a ring.

If (R,

+, )

is a commutative ring, a nonzero element a E R is called a zero divisor if there exists a nonzero element b ^E R such that ab = O. If a nontrivial

ring has no zero divisors, it is called an integral domain. In the same way that Z is extended to Q, an integral domain can be embedded in its field of fractions or quotient field.

(1.1.7) Definition. If (R,

+, )

is a ring and

0

i= S s; R, then S is called an ideal if

(i) VaeSVbeS[a - bE S], (ii) VaeSVbeR[abES /\ baES].

It is clear that if S is an ideal in R, then (S,

+, )

is a subring, but require- ment (ii) says more than that.

(1.1.8) Definition. A field is a ring (R,

+, )

for which (R \ {O}, ) is an abelian group.

(1.1.9) Theorem. Every finite ring R with at least two elements such that VaeRVbeR[ab = 0 =(a = 0 v b = 0)]

is a field.

(1.1.10) Definition. Let (V, +) be an abelian group, IF a field and let a multipli- cation IF x V -. V be defined satisfying

(i) V.ev[la = a],

V«efVpefV.ev[ex(pa) = (exp)a], (ii) V«efV.eVVbeV[ex(a

+

b)

=

exa

+

exb],

V«e FVpe FVaeV[(ex

+

p)a

=

exa

+

pal

Then the triple (V,

+,

IF) is called a vector space over the field IF. The identity element of (V, +) is denoted by

o.

We assume the reader to be familiar with the vector space IIlⁿ consisting of all n-tuples (aI' al , ... , an) with the obvious rules for addition and multiplica- tion. We remind him of the fact that a k-dimensional subspace C of this vector space is a vector space with a basis consisting of vectors a1 :=

(all' all' ... , a 1n), al := (all' all' ... , aln ), ... , ak := (akl' akl' ... , akn), where the word basis means that every a E C can be written in a unique way as exl a1

+

exlal

+ ... +

exkak • The reader should also be familiar with the process of going from one basis of C to another by taking combinations of basis vectors, etc. We shall usually write vectors as row vectors as we did above. The inner product <a, b) of two vectors a and b is defined by

<a, b) := al b1

+

al b2

+ ... +

anbn.

The elements of a basis are called linearly independent. In other words this means that a linear combination of these vectors is 0 iff all the coefficients are O. If a I , ••• , ak are k linearly independent vectors, i.e. a basis of a k-dimensional

§1.1. Algebra 5 subspace C, then the system of equations (ai' y) = 0 (i = 1, 2, ... , k) has as its solution all the vectors in a subspace of dimension n - k which we denote by Col. So,

Col := {y E IRnlV xeC[ (x, y) = O]}.

These ideas playa fundamental role later on, where IR is replaced by a finite field IF. The theory reviewed above goes through in that case.

(1.1.11) Definition. Let (V, +) be a vector space over IF and let a multiplica- tion V x V ~ V be defined that satisfies

(i) (V,

+, )

is a ring,

(ii) V"'eFV.eyVhy[(aa)b = a(ab)].

Then we say that the system is an algebra over IF.

Suppose we have a finite group (G, .) and we consider the elements of Gas basis vectors for a vecto,r space (V, +) over a field IF. Then the elements of V are represented by linear combinations algi

+

rt.2g2

+ ... +

_angn, where

ai Elf, (1 ~ i ~ n

=

IGJ).

We can define a multiplication

*

for these vectors in the obvious way, namely

which can be written as Lk Ykgk' where Yk is the sum of the elements aiPj over all pairs (i, j) such that gi' gj

=

gk' This yields an algebra which is called the group algebra of G over IF and denoted by IFG.

EXAMPLES. Let us consider a number of examples of the concepts defined above.

If A := {ai' a2, ... , an} is a finite set, we can consider all one-to-one map- pings of S onto S. These are called permutations. If a₁ and a2 are permutations we define al a2 by (a1 a2)(a) ^:= a₁ (a2(a» for all a E A. It is easy to see that the set Sn of all permutations of A with this multiplication is a group, known as the symmetric group of degree n. In this book we shall often be interested in special permutation groups. These are subgroups of Sn. We give one example.

Let C be a k-dimensional subspace of IRn. Consider all permutations a of the integers 1,2, ... , n such that for every vector c

=

(c_{1 ,} c_{2 ,} ••• , cn) E C the vector (ca(1)' Ca(2)' ••• , caIn)~ is also in C. These clearly form a subgroup of Sn. Of course C will often be such that this subgroup of S consists of the identity only but there are more interesting examples! Another example of a permutation group which will tum up later is the affine permutation group defined as follows. Let IF be a (finite) field. The mapping fu.v, when u E IF, v E IF, u :f= 0, is defined on IF by fu,v(x) := ux

+

^v for all x E IF. These mappings are permuta- tions of IF and clearly they form a group under composition of functions.

A permutation matrix P is a (0, I)-matrix that has exactly one 1 in each row and column. We say that P corresponds to the permutation (1 of {I, 2, ... , n}

if Pij = 1 iff i = (1(j) (i = 1,2, ... , n). With this convention the product of permutations corresponds to the product of their matrices. In this way one obtains the so-called matrix representation of a group of permutations.

A group G of permutations acting on a set

n

is called k-transitive on

n

if for every ordered k-tuple (al , ... , ak ) of distinct elements of

n

and for every k-tuple (bl , ... ,

1\)

of distinct elements of

n,

there is an element ⁽¹ E G such that bi ⁼ (1(ai) for 1 ~ i ~ k. If k

=

1 we call the group transitive.

Let S be an ideal in the ring (R,

+, ).

Since (S,

+)

is a subgroup of the abelian group (R, +), we can form the factor group. The cosets are now called residue classes mod S. For these classes we introduce a multiplication in the obvious way: (a

+

S)(b

+

S) := ap

+

S. The reader who is not familiar with this concept should check that this definition makes sense (i.e. it does not depend on the choice of representatives a resp. b). In this way we have constructed a ring, called the residue class ring R mod S and denoted by RjS.

The following example' will surely be familiar. Let R := 7l. and tet p be a prime.

Let S be p71., the set of all multiples of p, which is sometimes also denoted by (p). Then RjS is the ring of integers mod p. The elements of RjS can be represented by 0, 1, ... , p - 1 and then addition and mUltiplication are the usual operations in 7l. followed by a reduction mod p. For example, if we take p = 7, then 4

+

5 = 2 because in 7l. we have 4

+

5

=

^{2 (mod 7).} In the same way 4· 5 = 6 in 7l.j771. = 7l./(7). If S is an ideal in 7l. and S :/= {O}, then there is a smallest positive integer k in S. Let s E S. We can write s as ak

+

^b, where

o

^~ b < k. By the definition of ideal we have ak E S and hence b = s - ak E S and then the definition of k implies that b

=

O. Therefore S = (k). An ideal consisting of all multiples of a fixed element is called a principal ideal. If a ring R has no other ideals than principal ideals, it is called a principal ideal ring.

Therefore 71. is such a ring.

An ideal S is called a prime ideal if ab E S implies a E S or b E S. An ideal S in a ring R is called maximal if for every ideal J with S

c

J C R, J = S or J = R (S

i=

R). If a ring has a unique maximal ideal, it is called a local ring.

(1.1.12) Theorem. If p is a prime then 7l./p71. is a field.

This is an immediate consequence of Theorem 1.1.9 but also obvious directly. A finite field with n elements is denoted by IFn or GF(n) (Galois field).

Rings and Finite Fields

More about finite fields will follow below. First some more about rings and ideals. Let IF be a finite field. Consider the set IF[x] consisting of all polyno- mials ao

+

al x

+ ... +

anxn, where n can be any integer in Nand ai E IF for

o

~ i ~ n. With the usual definition of addition and multiplication of polyno-

§ 1.1. Algebra 7 mials this yields a ring (IF[xJ,

+, ),

which is usually just denoted by IF[x].

The set of all polynomials that are multiples of a fixed polynomial g(x), i.e. all polynomials of the form a(x)g(x) where a(x) E IF[xJ, is an ideal in IF[x].

As before, we denote this ideal by (g(x». The following theorem states that there are no other types.

(1.1.13) Theorem. IF [xJ is a principal ideal ring.

The residue class ring IF[xJ/(g(x)) can be represented by the polynomials whose degree is less than the degree of g(x). In the same way as our example 7l./T71. given above, we now multiply and add these representatives in the usual way and then reduce mod g(x). For example, we take IF = IFz ⁼ {O, I} and g(x)

=

x³

+

X

+

1. Then (x

+

l)(x2

+

1)

=

x³

+

x 2

+

X

+

1

=

x2. This ex- ample is a useful one to study carefully if one is not familiar with finite fields.

First observe that g(x) is irreducible, i.e., there do not exist polynomials a(x) and b(x) E IF[xJ, both of degree less than 3, such that g(x)

=

a(x)b(x). Next, realize that this means that in 1F2[xJ/(g(x)) the product of two elements a(x) and b(x) is 0 iff a(x)

=

0 or b(x)

=

O. By Theorem 1.1.9 this means that 1F2[xJ/(g{x» is a field. Since the representatives of this residue class ring all have degrees less than 3, there are exactly eight of them. So we have found a field with eight elements, i.e. 1F23 • This is an example of the way in which finite fields are constructed.

(1.1.14) Theorem. Let p be a prime and let g(x) be an irreducible polynomial of degree r in the ring IFp[x]. Then the residue class ring IFp[xJ/(g(x» is a field with pr elements.

PROOF. The proof is the same as the one given for the example p = 2, r = 3,

g(x) = x³

+

X

+

1. . D

(1.1.15) Theorem. Let IF be afield with n elements. Then n is a power of a prime.

PROOF. By definition there is an identity element for multiplication in IF. We denote this by 1. Of course 1

+

1 ^E IF and we denote this element by 2. We continue in this way, i.e. 2

+

1

=

3, etc. After a finite number of steps we encounter a field element that already has a name. Suppose, e.g. that the sum of k terms 1 is equal to the sum of I terms I (k > I). Then the sum of (k - I) terms I is 0, i.e. the first time we encounter an element that already has a name, this element is O. Say 0 is the sum of k terms 1. If k is composite, k = ab, then the product of the elements which we have called a resp. b is 0, a contradiction. So k is a prime and we have shown that IFp is a subfield of IF.

We define linear independence of a set of elements of IF with respect to (coefficients from) IFp in the obvious way. Among all linearly independent subsets of IF let ^{Xl' x2 , ••• , xr } be one with the maximal number of elements.

If x is any element of IF then the elements x, Xl' x 2, ... , Xr are not linearly

independent, i.e. there are coefficients 0 i= (X, (Xl' ••• , (x, such that !XX

+

^{(Xl Xl}

+ ... +

^(X,Xr = 0 and hence ^X is a linear combination of ^Xl to ^X,. Since there are obviously p' distinct linear combinations of Xl to X, the proof is complete.

0

From the previous theorems we now know that a field with n elements exists iff n is a prime power, providing we can show that for every r ~ 1 there is an irreducible polynomial of degree r in IFp[x]. We shall prove this by calculating the number of such polynomials. Fix p and let I, denote the number of irreducible polynomials of degree r that are monic, i.e. the coeffi- cient of xr is 1. We claim that

(1.1.16) (1 - pZ)-l = _,=1

n

00 ^{(1 -}

^zT

^1,.

In order to see this, first observe that the coefficient of

z"

on the left-hand side is p", which is the number of monic polynomials of degree n with coefficients in IFp. We know that each such polynomial can be factored uniquely into irreducible factors and we must therefore convince ourselves that these prod- ucts are counted on the right-hand side of (1.1.16). To show this we just consider two irreducible polynomials al (x) of degree rand a2(x) of degree s.

There is a 1-1 correspondence between products (al(x»k(a2(x»' and terms z~' z~ in the product of (1

+

z~

+ zf' + ... )

and (1

+

zi

+

z~s

+ "').

^If we identify z 1 and Z2 with z, then the exponent of z is the degree of (a 1 (x)t(a2(x»'.

Instead of two polynomials al (x) and a2(x), we now consider all irreducible polynomials and (1.1.16) follows.

In (1.1.16) we take logarithms on both sides, then differentiate, and finally mUltiply by

z

to obtain

(1.1.17) pz ⁰⁰ rz'

1 - pz =

,~

^{I, 1 _ z"}

Comparing coefficients of

z"

on both sides of (1.1) 7) we find

(1.1.18) p"

= L

rIr·

'1"

Now apply Theorem 1.l.4 to (1.1.18). We find

1 1

(1.1.19) I, = -

L

j.l(d)p,,4 > _ {p' - p,/2 _ p,/3 - ... }

r 41' r

1 ( ,,^{2 .)} 1

>- p'-

L

pi >_p'(1_p-,/2+1»0.

r ;=0 r

Now that we know for which values of n a field with n elements exists, we wish to know more about these fields. The structure of IF P' will playa very impor- tant role in many chapters of this book. As a preparation consider a finite field IF and a polynomial f(x) E IF[x] such that f(a) = 0, where a E IF. Then by dividing we find that there is a g(x) E IF[x] such that f(x) = (x - a)g(x).

§ 1.1. Algebra 9 Continuing in this way we establish the trivial fact that a polynomial f(x) of degree r in IF[x] has at most r zeros in IF.

If a is an element of order e in the multiplicative group (IF

p.\

{O}, ), then a is a zero of the polynomial xl! - 1. In fact, we have

x" - 1 = (x - l)(x - a)(x - (²)···(x - a"-I).

It follows that the only elements of order e in the group are the powers aⁱ where 1 ::;; i < e and (i, e) = 1. There are cp(e) such elements. Hence, for every e which divides p' - 1 there are either 0 or cp(e) elements of order e in the field.

By (1.1.1) the possibility 0 never occurs. As a consequence there are elements of order p' - 1, in fact exactly cp(p' - 1) such elements. We have proved the following theorem.

(1.1.20) Theorem. In IFq the multiplicative group (lFq \ {O}, ) is a cyclic group.

This group is often d~iloted by IF:.

(1.1.21) Definition. A generator of the multiplicative group of IFq is called a primitive element of the field.

Note that Theorem 1.1.20 states that the elements of IFq are exactly the q distinct zeros of the polynomial x^{q -} x. An element

fJ

^{such that}

fJk

^{= 1 but}

fJl

^{-=F 1 for} 0 < I < k is called a primitive kth root of unity. Clearly a primitive element a of IFq is a primitive (q - l)th root of unity. If e divides q - 1 then a"

is a primitive

«q -

1)/e)th root of unity. Furthermore a consequence of Theorem 1.1.20 is that IF".. is a subfield of IF P' itT r divides s. Actually this statement could be slightly confusing to the reader. We have been suggesting by our notation that for a given q the field IFq is unique. This is indeed true. In fact this follows from (1.1.18). We have shown that for q = pft every element of IFq is a zero of some irreducible factor of x^q - x and from the remark above and Theorem 1.1.14 we see that this factor must have a degree r such that rln.

By (1.1.18) this means we have used all irreducible polynomials of degree r where rln. In other words, the product of these polynomials is x q - x. This establishes the fact that two fields IF and IF' of order q are isomorphic, i.e. there is a mapping cp: IF -+ IF' which is one-to-one and such that cp preserves addition and multiplication.

The following theorem is used very often in this book.

(1.1.22) Theorem. Let q = p' and 0 -=F f(x) E IFq[x].

(i) If a E IFqk and f(a)

=

^0, then f(a q)

= o.

(ii) Conversely: Let g(x) be a polynomial with coefficients in an extension field of IFq. If g(aq ) = 0 for every a for which g(a) = 0, then g(x) E IFq[x].

PROOF.

(i) By the binomial theorem we have (a

+

^b)p

=

a^P

+

^{bP because p divides}

(n

for 1 :::;:; k :::;:; p - 1. It follows that (a

+

^b)q

=

aq

+

^{bq. If f(x)}

=

L aix i then (f(x))q = La?(xq)i.

Because _ai E IFq we have a?

=

_ai. Substituting x

=

a we find f(a^{q )} = (f(aW = O.

(ii) We already know that in a suitable extension field of IFq the polynomial g(x) is a product of factors x - ai (all of degree 1, that is) and if x - ai is one of these factors, then x - a? is also one of them. If g(x) =

Lk=O

akxt then at is a symmetric function of the zeros

a

i and hence at =

at,

i.e.

at E IFq.

If a E IFq, where q

=

^pr, then the minimal polynomial of a over IFp is the irreducible polynomial f(x) E IFp[xJ such that f(a)

=

O. If a has order e then from Theorem 1.1.22 we know that this minimal polynomial is n7'=ol (x - aP'), where m is the smallest integer such that pm == 1 (mod e).

Sometimes we shall consider a field IF ^q with a fixed primitive element a. In that case we use mi (x) to denote the minimal polynomial of ai • An irreducible polynomial which is the minimal polynomial of a primitive element in the corre- sponding field is called a primitive polynomial. Such polynomials are the most convenient ones to use in the construction of Theorem 1.1.14. We give an example in detail.

(1.1.23) EXAMPLE. The polynomial X4

+

^x

+

1 is primitive over 1F2 . The field 1F2' is represented by polynomials of degree < 4. The polynomial x is a primitive element. Since we prefer to use the symbol x for other purposes, we call this primitive element a. Note that a⁴

+

^a

+

^{1 =} O. Every element in 1F24 is a linear combination of the elements 1, ct, a2 , and a3 • We get the following table for 1F2,. The reader should observe that this is the equivalent of a table of logarithms for the case of the field IR.

The representation on the right demonstrates again that 1F2' can be inter- preted as the vector space (1F2

t,

where {I, a, a2 , a3 } is the basis. The left-hand column is easiest for multiplication (add exponents, mod 15) and the right- hand column for addition (add vectors). It is now easy to check that

m1(x) = (x -a)(x _(²)(x - (⁴)(x _as) = X4

+

^X

+

^1,

m3(x) = (x - (³)(x - (⁶)(x - (¹²)(x - (9 ) = X4

+

^{x 3}

+

^x2

+

^X

+

^1,

ms(x) = (x - as)(x - alO ) = x²

+

^X

+

^1,

m7(x) = (x - ( 7)(x - (¹⁴)(x - ( 13 )(x - all) = X4

+

x 3

+

1, and the decomposition of X 16 - x into irreducible factors is

§1.1. Algebra 11

X l6 _ x = x(x - 1)(x²

+

X

+

1)(X4

+

X

+

¹⁾

X (x4

+

^Xl

+

1)(x4

+

^Xl

+

x²

+

X

+

^1).

Note that X4 - X = x(x - 1)(x²

+

X

+

1) corresponding to the elements 0, 1, as, a^lO which form the subfield 1F4

=

1F2[x]/(X2

+

^X

+

1). The polynomial m₃(x) is irreducible but not primitive.

Table of !F24

o

= (0 0 0 0)

1

=

1

=

(l 0 0 0)

I;(

=

^C( = (0 1 0 0)

1;(2 1;(2

=

(0 0 1 0)

1;(3 rx³

=

(0 0 0 1)

rx⁴

=

1

+

^C(

=

(1 1 0 0) rx^S rx

+

^!X2 = (0 1 1 0)

1;(6 !X2

+

^1;(3

=

(0 0 1 1)

ri.7 = 1

+

^!X

+

^C(l

=

(1 1 0 1) rx⁸

=

1 + ^!X2 = (1 0 1 0)

!X9 C(

+

rx³

=

(0 1 0 1)

1;(10

=

1

+

^C(

+

^!X2

=

(1 1 1 0)

C(II

=

^!X + ^1;(2 + ^C(3

=

(0 1 1 1)

1;(12

=

1

+

^C(

+

^!X2

+

^!Xl = (1 1 1 1) rx^l3

=

1

+

^!X2

+

^C(3

=

(1 0 1 1)

rx^l4

=

1 -I-!X3

=

(l 0 0 1)

The reader who is not familiar with finite fields should study (1.1.14) to (1.1.23) thoroughly and construct several examples such as !F9, 1F27 , 1F64 with the corresponding minimal polynomials, subfields, etc. For tables of finite fields see references [9] and [10].

Polynomials

We need a few more facts about polynomials. If f(x) ^E IFq[x] we can define the derivative f'(x) in a purely formal way by

The usual rules for differentiation of sums and products go through and one finds for instance that the derivative of (x - a)2 f(x) is 2(x - rx)f(x)

+

(x - rx)2f'(X). Therefore the following theorem is obvious.

(1.1.24) Theorem. If f(x) E IFq[xJ and CI. is a multiple zero of f(x) in some extension field of IFq, then CI. is also a zero of the derivative f'(x).

Note however, that if q = 2', then the second derivative of any polynomial in IFq[x] is identically O. This tells us nothing about the multiplicity of zeros

of the polynomial. In order to get complete analogy with the theory of polynomials over IR, we introduce the so-called Hasse derivative of a polyno- mial f(x) ^E IFq[x] by

(so the k-th Hasse derivative of x· is

G)

^{x.-k )'}

The reader should have no difficulty proving that a is a zero of f(x) with multiplicity k iff it is a zero of flil(x) for 0

s

i < k and not a zero of jlkl(x).

Another result to be used later is the fact that if f(x)

= U7=1

(x - aJ then rex)

= Lt=1

f(x)/(x -

aJ

The following theorem is well known.

(1.1.25) Theorem. If the polynomials a(x) and b(x) in IF[x] have greatest common divisor 1, then .there are polynomials p(x) and q(x) in IF[x] such that

a(x)p{x)

+

b(x)q(x) = 1.

PROOF. This is an immediate consequence of Theorem 1.1.13.

o

Although we know from (1.1.19) that irreducible polynomials of any degree r exist, it sometimes takes a lot of work to find one. The proof of (1.1.19) shows one way to do it. One starts with all possible polynomials of degree 1 and forms all reducible polynomials of degree 2. Any polynomial of degree 2 not in the list is irreducible. Then one proceeds in the obvious way to produce irreducible polynomials of degree 3, etc. In Section 9.2 we shall need irreduc- ible polynomials over 1F2 of arbitrarily high degree. The procedure sketched above is not satisfactory for that purpose. Instead, we proceed as follows.

(1.1.26) Lemma.

PROOF.

(i) For

f3 =

0 and

f3 =

1 the assertion is true.

(ii) Suppose 3¹11(2³⁶

+

1). Then from

(2³⁶⁺¹

+

1) = (2³⁶

+

1){(2³⁶

+

1)(2^{36 -} 2)

+

3}, it follows that if t 2:: 2, then 3¹+¹11 (2^{36+ I}

+

1).

(1.1.27) Lemma. If m is the order of 2 (mod 3'), then m

=

qJ(3')

=

2· 3'-1.

o

§l.l. Algebra \3 PROOF. If 2« == 1 (mod 3) then ^IX is even. Therefore m = 2s. Hence 2'

+

1 ==

°

^{(mod 31).} The result follows from Theorem 1.1.2 and Lemma 1.1.26. 0 (1.1.28) Theorem. Let m = 2.31- 1• Then

xm

+

^Xml2

+

1 is irreducible over 1F2 •

PROOF. Consider 1F2^m. In this field let ~ be a primitive (3¹)th root of unity.

The minimal polynomial of ~ then is, by Lemma 1.1.27 f(x) = (x - ~)(x -

e

^{2)(x - ~4) ... (x _} ~2,"-1),

a polynomial of degree m. Note that

x^{3 '}

+

1 = (1

+

x)(1

+

x

+

x²)(1

+

x³

+

x⁶)···(1

+

^{X 31 -1}

+

^{X 2 • 31 -1),}

a factorization which contains only one polynomial of degree m, so the last

factor must be f(x), i.e. it is irreducible. 0

Quadratic Residues

A consequence of the existence of a primitive element in any field IFq is that it is easy to determine the squares in the field. If q is even then every element is a square. If q is odd then IFq consists of 0, t(q - 1) nonzero squares and t(q - 1) nonsquares. The integers k with 1 ~ k ~ p - 1 which are squares in IFp are usually called quadratic residues (mod pl. By considering k E IFp as a power of a primitive element of this field, we see that k is a quadratic residue (mod p) iff ^k(P-l)/2 == 1 (mod pl. For the element p - 1 = -1 we find: -1 is a square in IFp iff p == 1 (mod 4). In Section 6.9 we need to know whether 2 is a square in IFp. To decide this question we consider the elements 1, 2, ... , (p - 1)/2 and let a be their product. Multiply each of the elements by 2 to obtain 2, 4, ... , p - 1. This sequence contains

UP -

^1)/4

J

factors which are factors of a and for any other factor k of a we see that - k is one of the even integers> (p - 1)/2. It follows that in IFp we have 2(p-1)/2 a = (_1)(P-lJ/2-l(P-IJ/4J a and since a -#

°

we see that 2 is a square iff

p -₂ 1

-lp -

₄ ¹

^J

is even, i.e. p ==

±

1 (mod 8).

The Trace

Let q

=

pro We define a mapping Tr: IFq .... lFp , which is called the trace, as follows.

(1.1.29) Definition. If ~ E IFq then

Tr{~):= ~

+

^~p

+

^~p2

+ ... +

~P'"-I.

(1.1.30) Theorem. The trace function has the following properties:

(i) For every ~ E IFq the trace Tr{~) is in IFp;

(ii) There are elements ~ E IFq such that Tr{~) =F 0;

(iii) Tr is a linear mapping.

PROOF.

(i) By definition (Tr{~»P = Tr(~).

(ii) The equation x

+

x^P

+ ... +

X"..-I = 0 cannot have q roots in IFq, (iii) Since (~

+

^{,,)P = ~p}

+

"p and for every a E IFp we have a^P = a, this is

obvious. 0

Of course the theorem implies that the trace takes every value p-l q times and we see that the polynomial x

+

x^P

+ ... +

X"..-I is a product of minimal polynomials (check this for Example 1.1.23).

Characters

Let (G, +) be a group and let (T, ) be the group of complex numbers with absolute value 1 with multiplication as operation. A character is a homo- morphism

x:

G - T, i.e.

(1.1.31)

From the definition it follows that X{O) = 1 for every character X. If X(g) = 1 for all 9 E G then X is called the principal character.

(1.1.32) Lemma. If X is a character for (G, +) then

L

^{X(g) =}

{I GI, if

X is the principal character,

geG 0, otherwise.

PROOF. Let It E G. Then

X(h)

L

^X(g)

= L

^X(h

+

g)

= L

^X{k).

geG geG keG

If X. is not the principal character we can choose h such that X(h) =F 1. 0

§1.2. Krawtchouk Polynomials

In this section we introduce a sequence of polynomials which play an impor- tant role in several parts of coding theory, the so-called Krawtchouk polyno-

§1.2. Krawtchouk Polynomials 15 mials. These polynomials are an example of orthogonal polynomials and most of the theorems that we mention are special cases of general theorems that are valid for any sequence of orthogonal polynomials. The reader who does not know this very elegant part of analysis is recommended to consult one of the many textbooks about orthogonal polynomials (e.g. G. Szego [67], D. Jackson [36], F. G. Tricomi [70]). In fact, for some of the proofs of theorems that we mention below, we refer the reader to the literature. Because of the great importance of these polynomials in the sequel, we treat them more extensively than most other subjects in this introduction.

Usually the Krawtchouk polynomials will appear in situations where two parameters nand q have already been fixed. These are usually omitted in the notation for the polynomials.

(1.2.1) Definition. For k = 0, 1,2, ... , we define the Krawtchouk polynomial K,,(x) by

" .(x)(n - x) . K,,(x; n, q):= K,,(x):=

L

^{(-1)1 . k . (q - 1),,-J,}

j=O J - J

where

( x):= x(x - 1)· ··(x - j

+

¹⁾

j j! ' ^{(x E} IR).

Observe that for the special case q = 2 we have

(1.2.2) K,,(x) =

± ( - ^lY(~)

^(nk

^-~)

^{= ( -} 1)"K,,(n - x).

j=O J - J

By mUltiplying the Taylor series for (1

+

(q - l)zrX and (1 - z)X we find (1.2.3)

L

00 K,,(x)z"

=

⁽¹

+

(q - l)zrx(l - z)".

"=0

It is clear from (1.2.1) that K,,(x) is a polynomial of degree k in x with leading coefficient (-q)"/k! The name orthogonal polynomial is connected with the following "orthogonality relation":

(1.2.4)

it ^(~)<q

^{- l)iK,,(i)K1(i) =}

«5"I(~)(q

^- 1)"qn.

The reader can easily prove this relation by multiplying both sides by x"

yI

and summing over k and I (0 to (0), using (1.2.3). Since the two sums are equal, the assertion is true. From (1.2.1) we find

(1.2.5) (q -

1)(~)K"(i)

^{= (q -}

1)"(~)Ki(k)'

which we substitute in (1.2.4) to find a second kind of orthogonality relation:

n

(1.2.6)

;=0 2:

K,{i)K;(k) = ~'kqn.

We list a few of the Krawtchouk polynomials (k ~ 2) (1.2.7) Ko(n. x) = I,

K1(n, x) = n(q - 1) - qx, (= n - 2x if q = 2),

In Chapter 7 we shall need the coefficients of Xk, Xk-l, Xk-2, and XO in the expression of Kk(x). If Kk(x) =

2:7=0

^{CiXi. then for q =} 2 we have:

(1.2.8) ck

=

(-2)k/k!.

Ck-l

=

C-2)k-1n/(k - I)!.

Ck-2

= i(

_2)k-2{3n2 - 3n

+

2k - 4}/(k - 2)!'

For several purposes we need certain recurrence relations for the Krawt- chouk polynomials. The most important one is

(k

+

I )Kk+1 (x)

(1.2.9) = {k

+

(q - I)(n - k) - qx}Kk(x) - (q - I)(n - k

+

I)Kk_1(X).

This is easily proved by dilTerentiating both sides of (1.2.3) with respect to z and mUltiplying the result by (1

+

(q - I)z)(l - z). Comparison of coeffi- cients yields the result. An even easier exercise is replacing x by x-I in (1.2.3) to obtain

(1.2.10)

which is an easy way to calculate the numbers Kk(i) recursively.

If P(x) is any polynomial of degree I then there is a unique expansion (1.2.11) P(x) =

2:

I I'tkKk(x),

k=O

which is called the Krawtchouk expansion of P(x).

We mention without proof a few properties that we need later. They are special cases of general theorems on orthogonal polynomials. The first is the ChristolTel-Darboux formula

(1.2.12) Kk+t<x)Kk(y) - Kk(x)Kk+l(Y) = _2_(n)

±

K;(x)Ki(y).

Y - x k

+

1 k i=O

C)

The recurrence relation (1.2.9) and an induction argument show the very important interlacing property of the zeros of Kk(x):