Math 342
Homework Assignment #2 (Due Thursday, February
4, in class)
All numbered problems are worth the same value.
1. Let C be a code with minimum distance d(C) = 5.
(a) Show that the following hybrid decoder is guaranteed to correct 1 error and detect 2 errors and detect 3 errors:
Let x be the received word and c be a codeword which is a
nearest neighbour to x.
If d(x, c) ≤ 1, decode x to c.
If d(x, c) ≥ 2, declare an error.
(note that this is the same hybrid decoder that we described
for d(C) = 4).
(b) Let C be the binary 5-repetition code. For each of the received words x = 00000, 10000, 11000, 11100, 11110, 11111,
say what each of the following decoders will return (i.e.,
which codeword it will decode to or if it will declare an error)
i. An incomplete nearest neighbour decoder which corrects
up to 2 errors.
ii. A decoder which detects up to 4 errors.
iii. The hybrid decoder described in part a.
(c) Does there exist a hybrid decoder for the 5-repetition code
that is guaranteed to correct 1 error and detect 2 errors and
detect 3 errors and detect 4 errors. Why or why not?
2. Which of the following pairs of codes are equivalent? For each
pair, say why or why not?
(a) K1 = {00000, 11111}, K2 = {10100, 01011},
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(b) K1 = {000, 011, 101, 110}, K2 = {000, 011, 101, 100}.
(c) K1 = {000, 100, 010}, K2 = {000, 100, 111}.
3. Prove the following lower bound:
Aq (n, d) ≥ d Pd−1
m=0
qn
e.
n
m
m (q − 1)
Hints:
(a) Let C be a code which achieves Aq (n, d). Show that for
every word x ∈ Fqn, there is a codeword c ∈ C such that
d(x, c) < d.
(b) Show that Fqn = ∪x∈C Bd−1(x).
4. Do there exist codes with the following parameters? Why or
why not?
Hint: You may want to use the Singleton bound, Hamming
bound and the bound in problem 3.
(a) q = 2, (17, 800, 3)
(b) q = 2, (17, 8000, 3)
(c) q = 8, (5, 600, 3)
(d) q = 8, (5, 60, 3)
5. Which of the following subsets H of the integers Z, with the
usual addition as group operation, is a subgroup of Z? Why or
why not?
(a) H = {x ∈ Z : x is a multiple of 11}
(b) H = {x ∈ Z : x is a multiple of 22 and 33}
(c) H = {x ∈ Z : x − 6 is divisible by 11}
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6. Which of the following subsets H of Z23, with coordinate-wise
addition in the group Z3 as group operation, is a subgroup of
Z23? Why or why not?
Note: a ≡ b mod c means that c divides a − b.
(a) {(x, y) ∈ Z23 : x + 2y ≡ 0 mod 3}
(b) {(x, y) ∈ Z23 : x + 2y ≡ 1 mod 3}
(c) {(x, y) ∈ Z23 : x + 2y ≡ 0 mod 3 AND x + y ≡ 0 mod 3}
(d) {(x, y) ∈ Z23 : x + 2y ≡ 0 mod 3 OR x + y ≡ 0 mod 3}
7. Let G = Z52 and H = C3 (the code given on page 4 of the text).
Find all the cosets of H in G.
8. For each of the following sets S, verify that S is a subgroup and
calculate the number of cosets.
(a) S = the code {000000, 110110, 011011, 101101}, as a subgroup of Z62, with coordinate-wise addition.
(b) S = {00, 12, 21}, as a subgroup of Z23, with coordinate-wise
addition.
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