Math 447
Solutions to Homework 1
Fall 1998
Luis Goddyn
Page 17.
1.
(a) The rate of the Mariner code is k/n = 6/32 = 3/16.
(b) The rate of the Voyager code is k/n = 12/24 = 1/2.
2.
(a) d(0142, 3132) = 2.
(b) d(aabca, cabcc) = 2.
(c) d(, ) = 5.
4. Let a = (
MATH447/747 ASSIGNMENT 3
FALL 2012
The not to hand in questions
#13 (a) No column of H is 0 and no column in a multiple of another, while column 1
plus column 2 equals column 3, so the distance of this code is 3 and thus e = 1.
n = 4 and we are over Z3 an
Solutions to Homework 2
Fall 1998
Math 447
Luis Goddyn
Page 42.
2.
(a) 2.
(b) 2.
(c) 29.
(d) 3.
(e) 5.
3.
(a) Yes, since 2 is prime.
(b) No, since 2*2=0 in Z4 .
(c) Yes, since 7 is prime.
(d) No, since 3*3=0 in Z9 .
4. (a) Multiplying longhand and reducin
Math 447
Solutions to Homework 3
Fall 1998
Luis Goddyn
Page 95.
3.
(a) The code has 24 = 16 codewords.
(b) By Gauss Elimination we obtain a
1
0
0
0
generator matrix
0 0 0 1 1
1 0 0 1 1
.
0 1 0 1 1
0 0 1 0 1
This has the form [I|A]. A parity check matr
Math 447
Solutions to Homework 6
Fall 1998
Luis Goddyn
Page 261.
1. Yes. Just compare Examples 6 and 7 on page 37 of the textbook. In Example 6 we can get
a cyclic code generated by g(x) = m1 (x) = 1 + x + x4 , whereas in the Example 7 we have
m1 (x) = 1
MATH447/747 ASSIGNMENT 4
FALL 2012
initial questions
(1) (a) Fix a coordinate i. Let C (i) be the set of codewords from C which are 0 in
position i.
C (i) is closed under addition and scalar multiplication since 0 + 0 = 0 and c0 = 0
takes care of coordina
Constructing and computing in GF(81)
To construct GF(81) =
degree 4 over
[x]/f(x), we first find an irreducible polynomial f(x) of
.
We try some random ones until we get one. Each execution of the following should
produce a new candidate.
>
f:=x^4 + randp
First Midterm Exam
MATH 447-4
Family Name: SOLUTION KEY
Instructions: Do all ve questions, writing each answer in the space provided (use back if necessary).
Their point values are as indicated. Total = 50 points. Duration = 50 minutes.
1. Let C be an [n,
Here are examples of calculations on a,b,c over the rationals.
By default, all most "lower case" commands operate over the rationals.
>
a*b;
>
expand(a*b);expand(a^3);
>
factor(a); factor(b); factor(c);
>
irreduc(a); irreduc(b);
>
gcd(a,b); gcd(a,c);
Here
MATH447/747 ASSIGNMENT 1 SOLUTIONS
FALL 2012
(1) Many answers possible; nding all the parameters wasnt necessary, better to have an
interesting example.
3
6
(2) (a) The rate is 32 = 16 .
12
1
(b) The rate is 24 = 2 .
(3) (a) Let the codewords be cfw_c1 ,
Math 447
Solutions to Homework 7
Fall 1998
Luis Goddyn
Page 261.
1. In each case we list the possible values of n, which are simply the divisors of q 1. For each
such n the value k may be any value in the range 1, 2, . . . , n 1, and the resulting RS
code
MATH447/747 ASSIGNMENT 5 SOLUTIONS
FALL 2012
Not to hand in questions
Let c be a row of G. Then c c = 0 since c has weight divisible by 4. Along with the
given fact that the rows of G are pairwise orthogonal we get GGT = 0 and so C is
self-orthogonal.
Ta