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, 29 September 2022, 2:03 PM
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Completed on Thursday, 29 September 2022, 2:07 PM
Time taken 3 mins 32 secs
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Question 1
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There may be an error in the fifth digit in the following ISBN: 3-824-97290-8.
The value of the correct digit is:
Your last answer was interpreted as follows: 11
Incorrect answer.
Your proposed ISBN 3-824-17290-8 has syndrome 11, not zero as required.
ISBN-10 codes satisfy the check condition
∑i=110ixi≡0(mod11)∑i=110ixi≡0(mod11) .
A correct answer is 33, which can be typed in as follows:
Question 2
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You are given the following 7-bit ASCII codewords:
M 1001101 o 1101111 r 1110010 e 1100101
* 0101010 ^ 1011110 ‘ 0100111 5 0110101
Define a 5-character 8-bit ASCII code by encoding characters in blocks of 4 together with a 5th character which is used as a check codeword. (This is similar to the 9-character 8-bit ASCII code studied in lectures).
The message “More” together with its check character is given by:
(No answer given)
Incorrect answer.
To encode with this sort of code, we need to add check bits to the left of each codeword and also below each column in the grid of codewords. Those check bits below form the check character which, together with the codewords and their lefthand check bits would be transmitted as one long string of 0s and 1s, here 40 bits long. Here, however, we ignore the bits and just look at the characters that they represent, given in the table.
A correct answer is:
Question 3
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In a ternary linear code CC, the codeword x=x= 122022202201 has minimal weight among nonzero codewords.
What is the maximal numbers of errors that can always be corrected by CC?
Incorrect answer.
Hint: What is the minimum distance d(C)d(C) and how does it determine the code’s error detecting and correcting capabilities?
A correct answer is 44, which can be typed in as follows:
Question 4
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Let CC be a binary linear code with parity check matrix
H=⎡⎣⎢100100100010011001111⎤⎦⎥H=[111000100011010000111]
Assume that the check bits correspond to columns [1,4,5][1,4,5].
The codeword xx encoding the message m=m= 1011 in CC is:
Your last answer was interpreted as follows: 11
Incorrect answer.
Hint: Make sure that you haven’t confused the check bits and information bits.
A correct answer is 01010110101011, which can be typed in as follows:
Question 5
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Let CC be a binary linear code with check matrix
H=H=⎡⎣⎢⎢⎢010101000011001000011101⎤⎦⎥⎥⎥[000001110001001100101011]
with information bits in positions 2, and 5.
A message mm is encoded to a codeword xx.
This message is sent and received as the word y=y= 100010.
Assuming that there is at most one error, correct and decode yy to find the message mm:
Incorrect answer.
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 1111, which can be typed in as follows:
Question 6
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Let CC be the binary linear code with generator matrix
G=G= ⎡⎣⎢101101011011110010010⎤⎦⎥[110010000111111111000]
with information bits in positions 22, 44, and 66.
State the codeword xx that encodes the message m=m= 100:
Your last answer was interpreted as follows: 11221122
Incorrect answer.
Tip: Generator matrices are in general easier to encode with than parity check matrices (but harder to correct with) – but to get that benefit, you might first have to pivot the information bits in GG by row-reduction. That is probably overkill in this question, since the matrices are small.
A correct answer is 11001001100100, which can be typed in as follows:
Question 7
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Let CC be the ternary linear code with parity check matrix
H=H= ⎡⎣⎢⎢⎢01020101010000120011001000011220⎤⎦⎥⎥⎥[00000001111000020001110221021010]
and let BB be a basis for CC. How many codewords does BB contain?
Incorrect answer.
Hint: What is a basis?
A correct answer is 44, 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
4x1x1++5x24x2+4×3+5×4≡≡0(mod7)0(mod7)4×1+5×2+5×4≡0(mod7)x1+4×2+4×3≡0(mod7)
Assuming that x1x1 and x2x2 are the information bits,
find the codeword xx that encodes the message m=35m=35 :
Your last answer was interpreted as follows: 2121
Incorrect answer.
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 35313531, which can be typed in as follows:
Question 9
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Let CC be a radix 33 22-error correcting linear code with m=m= 44 check bits.
What is the greatest possible length nn of CC?
Your last answer was interpreted as follows: 22
Incorrect answer.
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 66, which can be typed in as follows:
Question 10
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Consider a binary channel with bit-error probability pp, where errors in different positions are independent (that is, white noise). Suppose that a codeword xx is sent from the binary repetition code with codewords of length 66, and the word yy is received with at least one error.
The probability that the error(s) in yy can be correctly corrected using the minimum distance decoding strategy is:
(No answer given)
1−(1−p)61−(1−p)6
15⋅(1−p)4⋅p215⋅(1−p)4⋅p2
20⋅(1−p)3⋅p3+15⋅(1−p)4⋅p2+6⋅(1−p)5⋅p20⋅(1−p)3⋅p3+15⋅(1−p)4⋅p2+6⋅(1−p)5⋅p
15⋅(1−p)4⋅p2+6⋅(1−p)5⋅p15⋅(1−p)4⋅p2+6⋅(1−p)5⋅p
15⋅(1−p)4⋅p2+6⋅(1−p)5⋅p+(1−p)615⋅(1−p)4⋅p2+6⋅(1−p)5⋅p+(1−p)6
Your last answer was interpreted as follows: 15⋅(1−p)4⋅p215⋅(1−p)4⋅p2
Incorrect answer.
Note: This question mostly just asks whether the code can correct given numbers of errors.
A correct answer is:
15⋅(1−p)4⋅p2+6⋅(1−p)5⋅p15⋅(1−p)4⋅p2+6⋅(1−p)5⋅p
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