Department of Mathematics
University of British Columbia
MATH 342 Practice Midterm 1
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100
CALCULATORS, NOTES OR BOOKS ARE NOT PERMITTED.
THERE ARE 5 PROBLEMS ON THIS EXAM.
JUSTIFY YOUR ANSWERS.
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1.
Let C be a code with minimum distance d(C) = 7.
(a) Find the maximum number of errors that can be corrected, using an
incomplete nearest neighbour decoder.
(b) Find the maximum number of errors that can be detected.
(c) Explicitly describe a hybrid decoder which is guaranteed to correct
up to 2 errors and to detect 3 errors and to detect 4 errors. Prove
that your decoder works.
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2.
True or False (if True, give a brief proof; if False, give a counter-example)
a) Every binary code C of length 8 is equivalent to a binary code which
contains the codeword 01010101.
b) Every binary code C of length 8 with d(C) = 8 is equivalent to a
binary code which contains the codewords 00000000 and 11111111.
c) If two binary codes have the same n, M, and d parameters, then they
are equivalent.
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3.
Find A2(10, 7).
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4. Let G = {I, R1, R2, F1, F2, F3} be the group of rigid motions that preserve an equilateral triangle centered at the origin, with composition of
mappings as addition. Here, I is the identity map, R1 is rotation by
2π/3 counter-clockwise, R2 is rotation by 4π/3 counter-clockwise, and
Fi, i = 1, 2, 3 is the reflection that fixes vertex i (with the vertices in
counter-clockwise order).
Recall that G is non-abelian.
(a) Show that H = {I, F1} is a subgroup of G.
(b) Find all the cosets, a + H, of H in G.
(c) Show that K = {I, R1, R2} is a subgroup of G.
(d) Find all the cosets, a + K, of K in G.
(e) Find the smallest subgroup of G that contains both F1 and F2.
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5.
Let q and m be positive integers. Show that Zm, as a group with addition
modulo m, has a subgroup of size q iff q divides m.
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