Department of Mathematics
University of British Columbia
MATH 342 Practice Midterm 2
Family Name:
Initials:
I.D. Number:
Problem
Signature:
Mark
Out of
1
20
2
20
3
20
4
20
5
20
Total
100
CALCULATORS, NOTES OR BOOKS ARE NOT PERMITTED.
THERE ARE 5 PROBLEMS ON THIS EXAM.
JUSTIFY YOUR ANSWERS.
MATH 342
Math 342 Practice Midterm 2
Below are the addition and multiplication tables for GF (4).
(GF (4), +):
+
0
1
a
b
0
0
1
a
b
1
1
0
b
a
a
a
b
0
1
b
b
a
1
0
·
0
1
a
b
0
0
0
0
0
1
0
1
a
b
a
0
a
b
1
b
0
b
1
a
(GF (4), ·):
2
MATH 342
Math 342 Practice Midterm 2
3
1. Compute the principal remainders of the following.
(a) (263999)20111 mod 5
(b) (735)24 mod 13
(c) (728)24 mod 13
2. A perfect square in a ring R is an element x ∈ R such that there exists y ∈ R such that x = y 2 .
A perfect cube in a ring R is an element x ∈ R such that there exists y ∈ R such that x = y 3 .
Find all the perfect squares and perfect cubes in the following rings.
(a) Z5
(b) Z6
(c) GF (4)
3. Let C be the span of the set of words {21211, 02010, 11102} in V (5, 3). Find the following.
(a) A generator matrix for C in standard form (or for a code equivalent to C by a permutation
of codeword positions)
(b) The dimension of C.
(c) |C|.
(d) The number of cosets of C (as a subgroup under vector addition) in V (5, 3).
4. Let n ≥ 2 be a positive integer. Let C be the set of all binary words of length n with even weight.
(a) Verify that C is a linear code over GF (2).
(b) Find the dimension of C in terms of n.
5. Let C be a linear ternary code of length n. Let D be the subset of C consisting of all x =
x1 . . . xn ∈ C such that x1 + x2 + · · · xn = 0 mod 3.
a) Show that D is a linear code.
b) Show that either |D| = |C| or |D| = |C|/3.