Midterm - People.stat.sfu.ca - Simon Fraser University

SIMON FRASER UNIVERSITY
DEPARTMENT OF MATHEMATICS
Midterm
Math 447/747 Fall 2012
Instructor: Dr. Yeats
October 12, 2012
Name: (please print)
family name given name
SFU email: @sfu.ca
SFU-email
Signature:
Instructions:
(1) Fill in your information above.
(2) Students taking 447 should complete any 5 questions. Students
taking 747 should complete any 6 questions. Please indicate which
questions you would like marked:
If you do not indicate anything the first answered ones will be marked.
(3) Answer in the spaces provided; use the back if necessary. Justify your answers.
(4) No calculators, books, papers, or electronic devices shall be within the reach
of a student during the examination.
(5) During the examination, communicating with, or deliberately exposing
written papers to the view of, other examinees is forbidden.

(1) (6 points) Prove algebraically that if C is an [n, M]-codeover the alphabet A with
distance d = 2e + 1 then C can correct e errors.
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(2) (6 points) Let C be an (n, k)-code with distance d. Let w(C) be the Hamming
weight of the code. Prove that d = w(C).
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(3) Let C be the binary linear code with parity check matrix
⎡
⎤
1 1 0 0 1
H = ⎣0 0 1 1 1⎦
1 0 1 0 0
(a) (2 points) What is the distance of C? Justify.
(b) (1 point) How many codewords does C have?
(c) (2 points) Give a generator matrix for this code.
(d) (1 point) Encode the following message words
(1 1) (0 1)
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(4) (a) (2 points) Give a parity check matrix for the binary Hamming code of order 3.
(b) (2 points) Decode the following received vectors
(1 0 1 1 0 1 1) (0 0 1 0 0 0 0)
(c) (2 points) Is your answer the only possible correct answer for part (a)? part
(b)? Explain.
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(5) Let C be the binary linear code with generator matrix
1 0 1 0 1
G =
0 1 0 1 1
(a) (4 points) Use any of the decoding algorithms in class to decode the following
received vectors (use the next page if you run out of space)
(1 1 1 0 1), (1 0 1 1 0), (0 1 0 1 1), (0 1 1 1 1), (0 1 0 0 0)
(b) (2 points) Suppose the message encoding was
O ↦→ 00 H ↦→ 10 L ↦→ 01 E ↦→ 11
What is the received message from the previous part?
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extra space if useful for the previous question
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(6) (4 points) Prove that in an (n, k)-code with distance d we must have
d ≤ n − k + 1.
(2 points) Give an example of a code which is perfect and has d = n − k + 1.
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(7) (6 points) Prove that an (n, k)-code C is self-dual if and only if n = 2k and C ⊆ C ⊥ .
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