Meeting
 Date
 Topic

1
 Wed, Jan 17, 2001
 Introduction

1.1 Basic problem

1.2 Three simple codes

2
 Fri, Jan 19, 2001
 1.3 Channel models

1.4 Encoders, error processors, decoders

1.5 Specific cases

Exercises: 4, 5, 6

3
 Mon, Jan 22, 2001
 2.1 Block codes

2.2 Weight and distance

2.3 Error processing

4
 Wed, Jan 24, 2001
 2.4 Error detection: necessary and sufficient conditions

2.5 Error correction: necessary and sufficient conditions

2.6 Mixed strategies

5
 Fri, Jan 26, 2001
 2.7 Probability of errors

2.8 Probability of correct transmission

2.9 Examples

2.10 Shannon's Theorem

Exercises: 1, 2, 4, 6, 7, 9, 10, 13, 14, 15

6
 Mon, Jan 29, 2001
 3.1 Arithmetic in B

3.2 Arithmetic in B^n

3.3 Fields: a definition

3.4 Vector space operations

7
 Wed, Jan 31, 2001
 3.5 Binary linear codes

3.6 Encoders and linearity

3.7 The generator matrix

8
 Fri, Feb 2, 2001
 3.8 The columns of the generator matrix

3.9 Codes and generator matrices

9
 Mon, Feb 5, 2001
 3.10 Code word checking

3.11 Relationship between generator and check matrices

3.12 Existence of generator and check matrices

3.13 Rank of the check matrix

Exercises: 6, 7, 10, 12, 16, 19, 21, 22, 23

10
 Wed, Feb 7, 2001
 4.1 Constructing the standard array

4.2 Row and column differences in a standard array

4.3 Occurrence of words in a standard array

4.4 Another view of Theorem 4.3

11
 Fri, Feb 9, 2001
 4.5 Cosets

4.6 Error words

4.7 Correction to closest code word

4.8 Information from the standard array

12
 Mon, Feb 12, 2001
 4.9 Cosets of a code and its check matrix

4.10 Syndromes of received words and error words

4.11 Condition for single error correction

4.12 Check matrix and minimum distance

Exercises: 1, 2, 4, 5, 6, 8

13
 Wed, Feb 14, 2001
 5.1 The binary Hamming codes

5.2 Parameters of the Hamming codes

5.3 Comparing Ham(3) and TPC

14
 Fri, Feb 16, 2001
 5.4 Perfect codes

5.5 Length of Hamming codes

5.6 A closer look at Ham(3)

5.7 Symmetries of the Hamming code

Exercises: 1, 2, 4, 5, 6

15
 Mon, Feb 19, 2001
 6.1 Constructing codes for correcting multiple errors

6.2 Correcting error bursts

6.3 Finding new codes

6.4 Fourbit strings

6.5 The integers modulo 16

6.6 Polynomials with binary coefficients

16
 Wed, Feb 21, 2001
 6.7 The structure of B[x]/f(x)

6.8 The field of order 16

Exercises: 4, 5, 6, 7, 8, 9, 10, 11

17
 Fri, Feb 23, 2001
 7.1 An example

7.2 Euclidean domains

7.3 Greatest common divisor

7.4 GCD in terms of a and b

18
 Mon, Feb 26, 2001
 7.5 The fourcolumn array for Euclid's algorithm

7.6 A worked example

7.7 A formal definition of Euclid's algorithm

7.8 More on Euclid's algorithm

Exercises: 1, 2, 3, 4, 8, 9

19
 Wed, Feb 28, 2001
 8.1 Invertible elements

8.2 More on GCDs

8.3 Invertibility and x

8.4 Relative primeness

20
 Fri, Mar 2, 2001
 8.5 The "1trick"

8.6 Irreducibility

8.7 The key property of irreducible elements

8.8 LCM of relatively prime elements

Exercises: 1, 2, 3, 7, 8, 9, 10, 11, 12, 13

21
 Mon, Mar 5, 2001
 9.1 The factor ring

9.2 The uniqueness assumption

9.3 D/a is a commutative ring

22
 Wed, Mar 7, 2001
 9.4 Remainder functions

9.5 Class representatives

9.6 Interchangeability of remainders

23
 Fri, Mar 9, 2001
 9.7 Condition for a field

9.8 Proof of the condition

9.9 Finding inverses

9.10 Available field sizes

9.11 GF(16) again

Exercises: 7, 8, 9, 10, 11, 12, 13

24
 Mon, Mar 12, 2001
 10.1 The prime field and the characteristic

10.2 Sizes of finite fields

10.3 A property of X(F)

10.4 Fermat's little theorem

25
 Wed, Mar 14, 2001
 10.5 Integer multiples

10.6 Some arithmetic

10.7 Constructing the prime field

10.8 Isomorphisms and automorphisms

10.9 Completing Theorem 10.1

10.10 Completing Theorem 10.2

26
 Fri, Mar 16, 2001
 10.11 Use of linear algebra

10.12 Uniqueness of the prime field

10.13 A result on binomial coefficents

10.14 The Frobenius automorphism

10.15 Termat's little theorem

Exercises: 1, 2, 3, 4, 5, 6, 7, 8

SPRING BREAK

27
 Mon, Mar 26, 2001
 11.1 More on polynomials

11.2 Evaluating polynomials

11.3 The formal derivative

11.4 Horner's scheme

11.5 The minimal polynomial of b

28
 Wed, Mar 28, 2001
 11.6 Properties of the minimal polynomial

11.7 Fields with p^n elements

11.8 The splitting field

11.9 An existence theorem

11.10 Herstein's alternative

11.11 Subfields of all orders

Exercises 1, 2, 3, 4, 5, 6, 7, 8

29
 Fri, Mar 30, 2001
 12.1 Primitive elements

12.2 Existence of primitive elements: preliminaries

12.3 Elements of large order

12.4 Existence of primitive element: proof

30
 Mon, Apr 2, 2001
 12.5 Discrete logarithms and addition

12.6 Primitive polynomials

12.7 Isomorphisms of fields of same order

12.8 Factorization of x^q  x

Exercises 1, 2, 3, 4, 5, 6, 7, 8

31
 Wed, Apr 4, 2001
 13.1 Introduction

13.2 Vandermonde matrices

13.3 Extending a Hamming check matrix

13.4 Verification

32
 Fri, Apr 6, 2001
 13.5 Further extensions

13.6 Using BCH(4, 3) as an example

13.7 List of code words

13.8 The reduced check matrix

33
 Mon, Apr 9, 2001
 13.9 Some questions

13.10 The check matrix and error patterns

Exercises: 1, 2, 3

14.1 Code polynomials

34
 Wed, Apr 11, 2001
 14.2 The generator polynomial

14.3 Rank and generator polynomial of BCH codes

14.4 Multiplicative encoding

35
 Fri, Apr 13, 2001
 14.5 A generator matrix for BCH(k, t)

14.6 The check polynomial

14.7 Multiplicative decoding for BCH(k, t)

36
 Mon, Apr 16, 2001
 14.8 Systematic encoding for BCH(k, t)

Exercises 1, 2, 3, 4, 5, 6, 7, 8, 9, 10

15.1 Determining the error word

15.2 The syndromes of a received word

37
 Wed, Apr 18, 2001
 15.3 Syndromes and syndrome vectors

15.4 The case s = 2

15.5 Summary of the method

38
 Fri, Apr 20, 2001
 15.6 The syndrome polynomial

15.7 A geometric progression

15.8 Formula for syndrome polynomial

39
 Mon, Apr 23, 2001
 15.9 Introduction to the fundamental equation

15.10 The numerators

15.11 The fundamental equation

15.12 Proving the fundamental equation

Exercises: 1, 2, 3, 4, 5, 6, 7

40
 Wed, Apr 25, 2001
 16.1 The fundamental equation again

16.2 The BCH algorithm

16.3 Termination of the algorithm

41
 Fri, Apr 27, 2001
 16.4 Failure modes

16.5 The polynomials calculated by the algorithm

16.6 Uniqueness of the error locator and evaluator

42
 Mon, Apr 30, 2001
 16.7 Properties of Euclid's algorithm

16.8 Relating l°, u°, w° to l, u, w

Exercises: 1, 2, 4, 5, 6
