108 CHAPTER 8. ALGEBRAIC CODING THEORY
PROOF. Two n-tuples x and y are in the same coset of C exactly when
x y E 0; however, this is equivalent to H(x y) = 0 or H: = Hy. El
Example 8.44. Table 8.42 is
9.2. DIRECT PRODUCTS 119
(1,5) + (c, d) = (a + c,b + d). The identity is (0,0) and the inverse of
(1,1) is (o, b).
Example 9.15. Consider
Z? X Z2 : cfw_(010)1(011)=(1:0)!(151)
Although Zg >< Z2 and Z4
8.5. EXERCISES 111
(C) (X130 = dfy: K);
(d) d(x.y) 5 d(x, 3) + OKAY)-
In other words, a metric is simply a generalization of the notion of dis-
tance. Prove that Hamming distance is a metric on ZS. De
8.4. EFFICIENT DECODING 105
Suppose now that we have a canonical parity-check matrix H with
three rows. Then we might ask how many more columns we can add to
the matrix and still have a null space tha
8. 7. REFERENCES AND SUGGESTED READINGS 1 13
Packing to Simple Groups. Carua Monograph Series, No. 21. Math-
ematical Association of America, Washington, DC, 1983.
[9] van Lint, J. H. Introduction to
A Note to the Student
This textbook presents an introduction to the theory and applications of complex
variables. The presentation has been molded by my belief that what you have
already studied in ca
Preface to the Second Edition
I was gratiﬁed by the very positive responses and reviews the ﬁrst edition of this
text received. Nonetheless, I wished to modify the text or exercises at a number of
poi
Preface to the First Edition
This textbook is intended for undergraduate or graduate students in science, mathe-
matics, and engineering who are taking their ﬁrst course in complex variables.
Its only
1.1 The Complex Numbers and the Complex Plane 3
Addition, subtraction, multiplication, and division of complex numbers follow
the ordinary rules of arithmetic. (Keep in mind that i2 = — 1, and, as usu
1
The Complex Plane
1.1 The Complex Numbers and the Complex Plane
The theory and utility of functions of a complex variable ultimately depend in large
measure on viewing the usual x- and y-coordinates
104 CHAPTER 8. ALGEBRAIC CODING THEORY
Theorem 8.31. Let H be an mxn binary matrix. Then the nail space of
H is a single error-detecting code if and only if no column of H consists
entirely of zeros.
106 CHAPTER 8. ALGEBRAIC CODING THEORY
PROOF. The proof follows from the fact that
Hx =H(c+e) = Hc+He =O+He = He.
[I
This proposition tells us that the syndrome of a received word depends
solely on th
112 CHAPTER 8. ALGEBRAIC CODING THEORY
(b) The column corresponding to the syndrome also marks the bit that
was in error; that is, the ith column of the matrix is 1' written as a
binary number, and th
102 CHAPTER 8. ALGEBRAIC CODING THEORY
Theorem 8.25. If H E men(Z2) is a canonical parity-check matrix,
then Null(H) consists of all x E Z; whose rst n m bits are arbitrary
but whose last in bits are
8.3. PARITYCHECK AND GENERATOR MATRICES 103
Theorem 8.28. Let H = (A \ Im) be an m x n canonical parity-check
matrix and let G : (him) be the n X (n in) standard generator matrix
associated with H.
118 CHAPTER 9. ISOMORPHISMS
The isomorphism g I:> A9 is known as the left regular representa-
tion of G.
|:| Historical Note I:|
Arthur Gayle,r was born in England in 1821, though he spent much
of the
9.1. DEFINITION AND EXAMPLES 117
Example 9.11. Consider the group 23. The Gayle:r table for Z3 is as
follows.
The addition table of Z3 suggests that it is the same as the permutaw
tion group G = cfw
116 CHAPTER 9. ISOMORPHISMS
PROOF. Assertions (1) and (2) follow from the fact that of) is a bijection.
We will prove (3) here and leave the remainder of the theorem to be
proved in the exercises.
(3)
9.1. DEFINITION AND EXAMPLES 115
Example 9.3. The integers are isomorphic to the subgroup of Q con-
sisting of elements of the form 2. Dene a map Q5 : Z > Q by em) = 2'.
Then
m + n) = 2W = 2W = was).
120 CHAPTER 9. ISOMORPHISMS
The next theorem tells us exactly when the direct product of two
cyclic groups is cyclic.
Theorem 9.21. The group Zm x 2, is isomorphic to Em if end. only
if gcd(rn,n) = 1.
110 CHAPTER 8. ALGEBRAIC CODING THEORY
(c) (d)
0001000
0110100
1110 1010010
1001 0110001
12. List all possible syndromes for the codes generated by each of the
matrices in Exercise 8.5.11.
13. Let
0
1
9.2. DIRECT PRODUCTS 121
Example 9.24. The group U(8) is the internal direct product of
H = cfw_1,3 and K = cfw_1,5.
Example 9.25. The dihedral group D5 is an internal direct product of
its two subgro
Isomorphisms
Man,r groups mar appear to be different at rst glance, but can be shown
to be the same by a simple renaming of the group elements. For example,
Z4 and the subgroup of the circle group '
COMPLEX VARIABLES
Second Edition DOVER BOOKS ON MATHEMATICS
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VECTORS, TENSORS AND THE BASIC EQUATIONS OF FLUID