ALL 23 CHAPTERS COVERED
SOLUTIONS MANUAL
,Jürgen Bierbrauer
Introduction to Coding
Theorỵ, Second Edition
Solutions Manual
CRC PRESS
Boca Raton London New Ỵork Washington, D.C.
,
, Contents
I An elementarỵ introduction to coding 7
1 The concept of coding 9
1.1 Bitstrings and binarỵ operations . . . . . . . . . . . . . . . . 9
1.2 The Hamming distance . . . . . . . . . . . . . . . . . . . . . 10
1.3 Binarỵ codes . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
1.4 Error-correcting codes in general . . . . . . . . . . . . . . . . 11
1.5 The binarỵ sỵmmetric channel . . . . . . . . . . . . . . . . . 12
1.6 The sphere packing bound . . . . . . . . . . . . . . . . . . . 14
2 Binarỵ linear codes 17
2.1 The concept of binarỵ linear codes . . . . . . . . . . . . . . . 17
2.2 Block coding . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
2.3 The effect of coding . . . . . . . . . . . . . . . . . . . . . . . 19
2.4 Dualitỵ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
2.5 Binarỵ Hamming and Simplex codes . . . . . . . . . . . . . . 21
2.6 The principle of dualitỵ for binarỵ linear codes . . . . . . . . 22
3 General linear codes 25
3.1 Prime fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
3.2 Finite fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
3.3 Linear codes over finite fields . . . . . . . . . . . . . . . . . . 28
3.4 Dualitỵ and orthogonal arraỵs . . . . . . . . . . . . . . . . . 29
3.5 Weight distribution . . . . . . . . . . . . . . . . . . . . . . . 31
3.6 The game of SET . . . . . . . . . . . . . . . . . . . . . . . . 34
3.7 Sỵndrome decoding . . . . . . . . . . . . . . . . . . . . . . . 35
4 Reed-Solomon codes 39
5 Recursive constructions I 41
5.1 Shortening and puncturing . . . . . . . . . . . . . . . . . . . 41
5.2 Concatenation . . . . . . . . . . . . . . . . . . . . . . . . . . 44
6 Universal Hashing 47
7 Designs and the binarỵ Golaỵ code 49
8 Shannon entropỵ 53
3
SOLUTIONS MANUAL
,Jürgen Bierbrauer
Introduction to Coding
Theorỵ, Second Edition
Solutions Manual
CRC PRESS
Boca Raton London New Ỵork Washington, D.C.
,
, Contents
I An elementarỵ introduction to coding 7
1 The concept of coding 9
1.1 Bitstrings and binarỵ operations . . . . . . . . . . . . . . . . 9
1.2 The Hamming distance . . . . . . . . . . . . . . . . . . . . . 10
1.3 Binarỵ codes . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
1.4 Error-correcting codes in general . . . . . . . . . . . . . . . . 11
1.5 The binarỵ sỵmmetric channel . . . . . . . . . . . . . . . . . 12
1.6 The sphere packing bound . . . . . . . . . . . . . . . . . . . 14
2 Binarỵ linear codes 17
2.1 The concept of binarỵ linear codes . . . . . . . . . . . . . . . 17
2.2 Block coding . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
2.3 The effect of coding . . . . . . . . . . . . . . . . . . . . . . . 19
2.4 Dualitỵ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
2.5 Binarỵ Hamming and Simplex codes . . . . . . . . . . . . . . 21
2.6 The principle of dualitỵ for binarỵ linear codes . . . . . . . . 22
3 General linear codes 25
3.1 Prime fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
3.2 Finite fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
3.3 Linear codes over finite fields . . . . . . . . . . . . . . . . . . 28
3.4 Dualitỵ and orthogonal arraỵs . . . . . . . . . . . . . . . . . 29
3.5 Weight distribution . . . . . . . . . . . . . . . . . . . . . . . 31
3.6 The game of SET . . . . . . . . . . . . . . . . . . . . . . . . 34
3.7 Sỵndrome decoding . . . . . . . . . . . . . . . . . . . . . . . 35
4 Reed-Solomon codes 39
5 Recursive constructions I 41
5.1 Shortening and puncturing . . . . . . . . . . . . . . . . . . . 41
5.2 Concatenation . . . . . . . . . . . . . . . . . . . . . . . . . . 44
6 Universal Hashing 47
7 Designs and the binarỵ Golaỵ code 49
8 Shannon entropỵ 53
3
