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 ninth digit in the following ISBN: 7-670-62730-6.
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 33, 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
10111000 11001110 10011111 11000000 00101101
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.
⎡⎣⎢⎢⎢⎢⎢⎢1111001010100011010011101001010110000101⎤⎦⎥⎥⎥⎥⎥⎥[1011100011001010100111111100000000101101]
A correct answer is:
Question 3
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Let CC be the binary linear code with parity check matrix
H=H=⎡⎣⎢000011011110110⎤⎦⎥[000110111101100]
What is the maximal numbers of errors that can always be detected, using the standard strategy, by CC?
Hint: First find the minimum distance d(C)d(C).
A correct answer is 00, 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=⎡⎣⎢⎢⎢100021000100111020102021⎤⎦⎥⎥⎥H=[120122011100000112000001]
Assume that the check bits correspond to columns [1,2,4,6][1,2,4,6].
The codeword xx encoding the message m=m= 11 in CC is:
Hint: Make sure that you haven’t confused the check bits and information bits.
A correct answer is 201210201210, 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=⎡⎣⎢⎢⎢0101010000120011001000011020⎤⎦⎥⎥⎥[0000001110000000111021021010]
with information bits in positions 1, 3, and 4.
A message mm is encoded to a codeword xx.
This message is sent and received as the word y=y= 1122210.
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 222222, 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= ⎡⎣⎢101110110101110001100⎤⎦⎥[111110101101001001010]
What are the dimensions of a (full rank) generator matrix GG for CC?
[Give your answer in the form axb if the matrix has dimension a×ba×b, e.g., “2×3”]
Hint: This question is very easy if you know the definitions of parity check matrices and generator matrices.
A correct answer is 4x74x7, which can be typed in as follows:
Question 7
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Consider the ternary linear code CC with basis
{01210,12112}{01210,12112}
What is the minimum distance d(C)d(C) of CC?
Hint: There are shortcuts that might be of use – but since the basis is small here, it might be easiest just to look at its codewords a little carefully.
A correct answer is 33, which can be typed in as follows:
Question 8
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Let CC be the code consisting of all vectors x=x1x2x3x4∈Z45x=x1x2x3x4∈Z54 satisfying the check equation
2x13x1+3×2+x3++x44x4≡≡0(mod5)0(mod5)2×1+x3+x4≡0(mod5)3×1+3×2+4×4≡0(mod5)
Assuming that x1x1 and x4x4 are the information bits,
find the codeword xx that encodes the message m=34m=34 :
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 30043004, which can be typed in as follows:
Question 9
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Consider a radix 33 (possibly non-linear) code CC of length n=5n=5 and minimum distance d(C)=1d(C)=1.
What is the greatest possible value of |C||C|, according to the Sphere-Packing Bound?
Tip: The code CC is not necessarily non-linear, so its size is not as fixed.
Note that a code CC 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 243243, which can be typed in as follows:
Question 10
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Consider a binary channel with probabilities
P(0 sent)=P(0 sent)= 0.20.2, P(1 received │0 sent)=P(1 received │0 sent)= 0.20.2, and P(0 received │1 sent)=P(0 received │1 sent)= 0.30.3 .
The probability P(0 received )P(0 received ) is
[Please provide your answer with two significant figures.]
Note: This is a simple question but we will see more interesting extensions of this question in Chapter 4.
A correct answer is 0.40.4, which can be typed in as follows:
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