MTH 416
Introduction to Algebraic Coding
F12
Homework 6
Justify all your answers
due on Wed 12/05/12
# 1. Let F be a eld, 0 = h F[x], f F[x] and g the canonical generator for the ideal f in Fh [x].
Show that
(a) g divides f and h in F[x].
(b) g = u
f for
MTH 416
Introduction to Algebraic Coding
F12
Homework 4/Solutions
Lemma A. (a) Let be an I J-channel and p a probability distribution on I. Then
H (; p) H ( p) ().
(b) Let C Bn be code with information rate . Let = BSCn (e) C Bn . If p is the equal-probab
MTH 416
Introduction to Algebraic Coding
F12
Homework 3/Solutions
# 1. Let S be an alphabet. For x S let qx be a probability distribution on S . For x S dene P (x)
inductively as follows:
P () = 1
and if P (x) already has been dened, then dene
P (xs) = P
MTH 416
Introduction to Algebraic Coding
F12
Practice Exam/Solutions
# 1. Construct a Human code for a source with alphabet S = cfw_a, b, c, d and probability distribution
p = (0.2, 0.2, 0.3, 0.3).
a
b
c
d
00
01
10
11
0.2
0.2
0.3
0.3
0
10
11
0.4
0.3
0.3
0
MTH 416
Fall 2012
Introduction to Algebraic Coding/Review For Final
#1. Construct a Human code for a source with alphabet S = cfw_1, 2, 3, 4, 5 and probability
distribution p = (0.1, 0.3, 0.15, 0.2, 0.25).
#2. Use the LZW-decoding rules to decode 352 with
MTH 416
Introduction to Algebraic Coding
F12
Homework 2/Solutions
# 1. What is the entropy to the base 2 of probability distribution (0.2, 0.3, 0.2, 0.2, 0.1)?
p
0.2
log2
p log2
1
p
0.2
0.1
5
5
10
1.7
2.3
2.3
3.3
0.46
0.2
10
3
2.3
1
p
0.3
5
1
p
0.51
0.46
MTH 416
Introduction to Algebraic Coding
F12
Solutions for Homework 1
# 1. The code ck from Example I.20 (or page 9 in the book) shifting each letter by k -places
is used to encode a message. If the encoded messages is
QY QBOOX QY GRSDO
what value for k w
MTH 416 Introduction to Algebraic Coding/ Practice Exam
10/14/15
Solutions
#1. Does there exist a uniquely decodable ternary code with parameter (0, 2, 2, 2, 2, 2)?
Yes: Let K be the Kraft-McMillan number of this parameter to the base 3. Then
5
K=
i=1
2
2
MTH 416
Introduction to Algebraic Coding
F5
Review for Exam 1/Solutions
During the exam you will be allowed to use the text book and the online
lecture notes, but not the solutions to the homeworks.
#1. Does there exist a prex-free ternary code with param
MTH 416
Introduction to Algebraic Coding
F15
Homework 1/Solutions
# 1. The code ck from Example I.21 (or page 9 in the book) shifting each letter by k-places
is used to encode a message. If the encoded messages is
QY QBOOX QY GRSDO
what value for k was us
MTH 416
Introduction to Algebraic Coding
F15
Homework 2
Solutions
# 1. Does there exist a prex-free ternary code with the following parameters:
a) n0 = 0, n1 = 1, n2 = 3, n3 = 10
(b) n0 = 0, n1 = 0, n2 = 1, n3 = 11, n4 = 39.
Let K be the corresponding Kra
Introduction to Algebraic Coding/ Exam 1
MTH 416
F2
Solutions
#1. Does there exist a prex-free code c cfw_A, B, C . . . , Z cfw_1, 2, 3, 4, 5 with parameters n1 = 3, n2 = 8 and
n3 = 15?
Since cfw_1, 2, 3, 4, 5 = 5, we compute the Kraft McMillan number K
MTH 416
Introduction to Algebraic Coding Practice Final
Fall 2012
Final Exam: Monday December 10, 12:45-2:45
You will be allowed to use the textbook and a print-out of the online lecture notes.
# 1. Construct a Human code for a source with alphabet S = cf
MTH 416
Introduction to Algebraic Coding Practice Final
Fall 2012
Final Exam: Monday December 10, 12:45-2:45
You will be allowed to use the textbook and a print-out of the online lecture notes.
# 1. Construct a Human code for a source with alphabet S = cf
MTH 416
Introduction to Algebraic Coding
F12
Practice Exam 1
Justify all your answers
The rst six exercise are from the exam I gave two years ago. The last two I had
considered, but didnt use. The exam had one more exercise, but it was from a section we
h
MTH 416
Introduction to Algebraic Coding
F12
Homework 5/Solutions
# 1. Let C be the binary linear code with check matrix
1
0
1
1
1
0
1
0
1
1
1
0
H=
0
0
1
1
(a) Find a syndrom look-up table for C with respect to H .
(b) Let be the decision rule correspondi
MTH 416
Introduction to Algebraic Coding Practice Final
Fall 2012
Final Exam: Monday December 10, 12:45-2:45
You will be allowed to use the textbook and a print-out of the online lecture notes.
# 1. Construct a Human code for a source with alphabet S = cf
MTH 416
Introduction to Algebraic Coding/ Exam 1
10/12/12
Justify all your answers
Name:
#1. Does there exist a prex-free code c : cfw_A, B, C . . . , Z cfw_1, 2, 3, 4, 5 with parameters n1 = 3,
n2 = 8 and n3 = 15.
#2. Let (S, P ) be a memoryless source
MTH 416
Introduction to Algebraic Coding/ Exam 1/Solutions
F15
#1. Does there exist a uniquely decodable binary code with parameter (0, 0, 0, 2, 4, 8, 17)?
No: The base for a binary code is b = 2. So we compute the Kraft-McMillan number K of the
given par