Practice Test 1: Attempt review
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MATH3411 Information, Codes and Ciphers (2022 T3)
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MATH3411-5229_00252
Practice Test 1
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Started on Thursday, 6 October 2022, 1:49 AM
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Completed on Thursday, 6 October 2022, 1:49 AM
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Question 1
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There may be an error in the fifth digit in the following ISBN: 4-085-32850-5.
The value of the correct digit is:
ISBN-10 codes satisfy the check condition
∑i=110ixi≡0(mod11)∑i=110ixi≡0(mod11) .
A correct answer is 55, which can be typed in as follows:
Question 2
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A message is sent using a 5-character 8-bit ASCII code which encodes characters in blocks of four together with a 5th character which is used as a check character for even parity in rows and columns, similar to the 9-character 8-bit ASCII code. The message
11110110 00100101 00100010 01011010 10101001
is received. Assuming at most one error, which of the following bits could be incorrect?
(No answer given)
Each row and column of the 5×85×8 grid must have an even number of 1s.
Those that don’t must contain error(s); this lets us find and sometimes correct possible errors.
The corrected grid for this question is shown below. As each row and column should have an even number of 1s, we can find the error, in the row and column that has an odd number of 1s.
⎡⎣⎢⎢⎢⎢⎢⎢1000110010111011001000011110001111001001⎤⎦⎥⎥⎥⎥⎥⎥[1111011000100111001000100101101010101001]
A correct answer is:
Question 3
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Let CC be the ternary linear code with parity check matrix
H=H=⎡⎣⎢010122021002110⎤⎦⎥[010011220102120]
What is the minimum distance d(C)d(C) of the code CC?
Hint: When we’re dealing with linear codes, there’s a nice shortcut to finding minimum weights and distances: see Tutorial Problem 19 and its solutions (written and on YouTube).
A correct answer is 33, which can be typed in as follows:
Question 4
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Let CC be a ternary linear code with parity check matrix
H=⎡⎣⎢⎢⎢10002000010020101010002001100001⎤⎦⎥⎥⎥H=[12021000001000100001121000000001]
Assume that the check bits correspond to columns [1,3,4,8][1,3,4,8].
The codeword xx encoding the message m=m= 0112 in CC is:
Hint: Make sure that you haven’t confused the check bits and information bits.
A correct answer is 0011112000111120, which can be typed in as follows:
Question 5
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Let CC be a ternary linear code with check matrix
H=H=⎡⎣⎢101100012010001022⎤⎦⎥[110000001102102012]
with information bits in positions 1, 4, and 5.
A message mm is encoded to a codeword xx.
This message is sent and received as the word y=y= 201001.
Assuming that there is at most one error, correct and decode yy to find the message mm:
Tip: Of the three operations encoding, correcting, and decoding, decoding is by far the easiest – once you have found the corrected codeword xx, just delete the check bit positions to find the message mm .
A correct answer is 200200, which can be typed in as follows:
Question 6
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Let CC be the binary linear code with parity check matrix
H=H= ⎡⎣⎢101011010001100⎤⎦⎥[100010110011010]
Which of the following is a generator matrix for CC?
(No answer given)
[0010101001][0111000001]
[1011110110][1110101110]
[0100000101][0000010011]
[1111110011][1110111101]
None of these
Tip: The rows of each generator matrix GG of a linear code CC are each codewords of CC .
A correct answer is:
[1011110110][1110101110]
Question 7
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Consider the binary linear code CC with basis
{0000001,0100000,1010110,1101010}{0000001,0100000,1010110,1101010}
What is the number of codewords in CC?
Hint: 1st year linear algebra is very useful in this course!
A correct answer is 1616, which can be typed in as follows:
Question 8
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Let CC be the code consisting of all vectors x=x1x2x3x4∈Z47x=x1x2x3x4∈Z74 satisfying the check equation
x15x2++x33x3++4x45x4≡≡0(mod7)0(mod7)5×2+x3+4×4≡0(mod7)x1+3×3+5×4≡0(mod7)
Assuming that x3x3 and x4x4 are the information bits,
find the codeword xx that encodes the message m=42m=42 :
Hint: Watch out that you don’t confuse the information bits with the check bits – that’s very easy to do.
A correct answer is 66426642, which can be typed in as follows:
Question 9
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Let CC be a radix 22 11-error correcting linear code with k=k= 22 information bits.
What is the smallest possible number of check bits mm in CC?
Hint: It is here useful to know that CC is linear.
The question would be quite different if CC were non-linear.
A correct answer is 33, which can be typed in as follows:
Question 10
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Consider a symmetric binary channel with constant bit-error probability pp, where errors in different positions are independent.
Suppose that a codeword xx is sent from the binary repetition code with codewords of length 44, and the word yy is received.
The probability that the error(s) in yy can be detected using a pure error detection strategy is:
(No answer given)
p4+4⋅(1−p)⋅p3+6⋅(1−p)2⋅p2+4⋅(1−p)3⋅pp4+4⋅(1−p)⋅p3+6⋅(1−p)2⋅p2+4⋅(1−p)3⋅p
4⋅(1−p)⋅p3+6⋅(1−p)2⋅p2+4⋅(1−p)3⋅p4⋅(1−p)⋅p3+6⋅(1−p)2⋅p2+4⋅(1−p)3⋅p
4⋅(1−p)⋅p3+6⋅(1−p)2⋅p2+4⋅(1−p)3⋅p+(1−p)44⋅(1−p)⋅p3+6⋅(1−p)2⋅p2+4⋅(1−p)3⋅p+(1−p)4
4⋅(1−p)3⋅p4⋅(1−p)3⋅p
4⋅(1−p)3⋅p+(1−p)44⋅(1−p)3⋅p+(1−p)4
Note: This question mostly just asks whether the code can detect given numbers of errors.
A correct answer is:
4⋅(1−p)⋅p3+6⋅(1−p)2⋅p2+4⋅(1−p)3⋅p4⋅(1−p)⋅p3+6⋅(1−p)2⋅p2+4⋅(1−p)3⋅p
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