Da… / M… / MATH3411-5249_00219 / Assessments Hub / Practice Tests 1-3 + BCH question (Chapters 1-6)
Started on Friday, 22 November 2024, 10:34 AM
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Completed on Friday, 22 November 2024, 10:35 AM
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Question 1
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QUIZ
Arithmetic coding is used with source symbols , and stop symbol represented by probabilities
, and
What can the message bab be encoded as?
(Please provide a valid message (number).)
 
a b ∙
0.6 0.2 0.2
∙
The message can be any number in the interval .
A correct answer is , which can be typed in as follows:
0.6936
[0.6912, 0.696)
0.6936

 
Question 2
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Question 3
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Let and . 
You may assume that is a eld.
A BCH code is obtained from by replacing a message
by the coecients of a polynomial where
and is the minimal polynomial of , where is a root of .
Using this code, encode the message :
Use the code to correct and decode the received message , assuming that at
most two errors occurred in transmission:
(x) = + x + 1 ∈ [x]m
1
x
4
Z
2
F = [x]/⟨ (x)⟩Z
2
m
1
F
F ( , ,… , )c
8
c
9
c
14
C(x) ∈ [x]Z
2
C(x)
(x)C
I
(x)C
R
m(x)
=
=
=
=
(x) + (x)C
R
C
I
+ +⋯+c
8
x
8
c
9
x
9
c
14
x
14
(x) mod m(x)C
I
(x) (x) = + + + + 1m
1
m
3
x
8
x
7
x
6
x
4
(x) = + + + x+ 1m
3
x
4
x
3
x
2
α
3
α (x)m
1
m = 0101100
y = 000010101000001
A correct answer is , which can be typed in as follows:
001010100101100
A correct answer is , which can be typed in as follows:
1001011
001010100101100
1001011
Calculate :
Then, using Euler's Theorem or otherwise, calculate :
 
ϕ(361)
mod 3612
361
A correct answer is , which can be typed in as follows:
342
A correct answer is , which can be typed in as follows:
116
342
116
 
Question 4
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Question 5
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Consider a radix -error correcting (possibly non-linear) code of length .
What is the greatest possible value of , according to the Sphere-Packing Bound?
3 2 C n = 8
|C|
Tip: The code is not necessarily non-linear, so its size is less xed.
Note that a code with the parameters that we have here might not actually exist - but we are
choosing to ignore this possibility for the purposes of this question.
A correct answer is , which can be typed in as follows:
50
C
C
50
Consider a binary channel with source symbols and output symbols  with
,        ,       
First nd ,  and  ; then use these to nd .

 
 
 
[Round your solutions to 2 decimal places]
{ , }a
1
a
2
{ , }b
1
b
2
P( ) =a
1
0.81 P( | ) =b
1
a
1
0.91 P( | ) =b
2
a
2
0.95
P( )a
2
P( )b
1
P( )b
2
H(B)
P( ) =a
2
P( ) =b
1
P( ) =b
2
H(B) =
A correct answer is , which can be typed in as follows:
0.19
A correct answer is , which can be typed in as follows:
0.75
A correct answer is , which can be typed in as follows:
0.25
A correct answer is , which can be typed in as follows:
0.82
0.19
0.75
0.25
0.82
 
Question 6
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Question 7
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A radix instantaneous code (I-code) has codeword lengths (not necessarily in order) 1,2,3,3, and
Kraft-McMillan coecient .
What is the value of ?
4 ℓ
K =
13
32
ℓ
Hint: The Kraft-McMillan Theorem is very useful, and its 3-part proof is worth studying and
understanding, even if it is probably the hardest of the (few slightly hard) proofs in the course.
A correct answer is , which can be typed in as follows:
2
2
Consider a symmetric binary channel with constant bit-error probability , where errors in different
positions are independent.
Suppose that a codeword is sent from the binary repetition code with codewords of length , and the
word is received.
The probability that there is at least one error and that the error(s) in  can be detected using a pure
error detection strategy is:
(No answer given)
 
 
 
p
x 5
y
y
+ 5 ⋅ (1 − p) ⋅ + 10 ⋅ ⋅ + 10 ⋅ ⋅ + 5 ⋅ ⋅ pp
5
p
4
(1 − p)
2
p
3
(1 − p)
3
p
2
(1 − p)
4
10 ⋅ ⋅ + 5 ⋅ ⋅ p(1 − p)
3
p
2
(1 − p)
4
5 ⋅ (1 − p) ⋅ + 10 ⋅ ⋅ + 10 ⋅ ⋅ + 5 ⋅ ⋅ pp
4
(1 − p)
2
p
3
(1 − p)
3
p
2
(1 − p)
4
10 ⋅ ⋅ + 5 ⋅ ⋅ p+(1 − p)
3
p
2
(1 − p)
4
(1 − p)
5
5 ⋅ (1 − p) ⋅ + 10 ⋅ ⋅ + 10 ⋅ ⋅ + 5 ⋅ ⋅ p+p
4
(1 − p)
2
p
3
(1 − p)
3
p
2
(1 − p)
4
(1 − p)
5
Note: This question mostly just asks whether the code can detect given numbers of errors.
A correct answer is:
5 ⋅ (1 − p) ⋅ + 10 ⋅ ⋅ + 10 ⋅ ⋅ + 5 ⋅ ⋅ pp
4
(1 − p)
2
p
3
(1 − p)
3
p
2
(1 − p)
4
 
Question 8
Not answered
Marked out of 1.00
Question 9
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A Markov source has transition matrix
 
State the Huffman code Huff for associated to and the 2nd column of :
[Please write your answer as a list enclosed by square brackets; e.g., [00,11].]
 
S = { , , }s
1
s
2
s
3
M =
⎡
⎣
⎢
⎢
3
5
3
10
1
10
3
5
3
10
1
10
1
10
1
5
7
10
⎤
⎦
⎥
⎥
2
S s
2
M
Hint: Remember to sort the symbols rst!
A correct answer is , which can be typed in as follows:
[0,10,11]
[0,10,11]
Using Fermat factorisation, factor into a product where .
What is the value of ?
n = 35581 n = ab 2 ≤ a < b
b− a
A correct answer is , which can be typed in as follows:
60
60
 
Question 10
Not answered
Marked out of 1.00
Let be the binary linear code with parity check matrix

and let be a basis for . How many codewords does contain?
C
H =
⎡
⎣
⎢
⎢
⎢
0
1
1
0
0
1
0
1
0
1
0
0
0
0
1
1
0
0
1
0
0
0
0
1
1
1
1
1
⎤
⎦
⎥
⎥
⎥
B C B
Hint: What is a basis?
A correct answer is , which can be typed in as follows:
3
3
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