MATH 220
Homework 1
Solutions
1. This question is about the code and transmission channel described in the lecture notes on page 4 (the codewords
are 000 and 111). Assuming that the codewords 000 and 111 are equally likely to be transmitted, prove that th
MATH 220
Homework 6
Solutions
1. It is given that f (x) = x4 + x + 1 is irreducible and primitive over GF (2). Dene by f () = 0.
(a) Set up a table for GF (16) containing the binary, exponential and polynomial representation of every element
in GF (16).
c
MATH 220
Homework 5
Solutions
1. Let C be a linear ternary code that is self-orthogonal (so C C ). Prove that the weight of any codeword is a
multiple of 3.
Solution : Note that 12 1 mod 3 and 22 4 1 mod 3. So if F := cfw_0, 1, 2 and = 0 then 2 = 1.
Let c
MATH 220
Homework 3
0
1. Let C be the binary code with generator matrix 1
1
0
1
0
1
0
1
Solutions
0
1
1
1
1 .
0
(a) Write down all the codewords in C.
(b) Find a generator matrix in standard form for C.
(c) Find a parity check matrix for C.
(d) Find all t
MATH 220
Homework 8
Solutions
1. Let n N and f (x) GF (q)[x]. Then f (x) generates a cyclic code C of length n over GF (q), namely
C = cfw_c GF (q)n : c(x) f (x)a(x)
mod (xn 1) for some a(x) GF (x)
Example : Let n = 3, q = 2 and f (x) = 1 + x2 . Then
C
mo
MATH 220
Homework 9
Solutions
1. How many cyclic codes of length 4 are there over GF (3)? Write down the generator polynomial for each of these
cyclic codes.
Solution : (a) The cyclotomic cosets depending on n = 4 and q = 3 are
cfw_0 , cfw_1, 3 and cfw_2
MATH 220
Exam 1: Part I
Solutions
1. TRUE/FALSE? Prove your answer!
(a) (5 pts) There exists a binary one-error correcting code of length 9 with 52 codewords.
(b) (5 pts) There exists a ternary one-error correcting code of length 6 with 10 codewords.
Solu
MATH 220
Homework 2
Solutions
1. Prove that there does not exist a binary one-error correcting code of length 6 with 9 codewords.
Proof : Suppose that C is a binary one-error correcting code of length 6 with 9 codewords. Then there are at
least ve codewor
MATH 220
Homework 4
1. Let C be the binary code word generator matrix
1
0
0
1
1
0
Solutions
0
1
.
(a) Set up a standard array for C.
(b) Write down the coset leaders of each coset.
(c) Use your standard array to decode the following words :
1111
1110
1
MATH 220
Homework 7
Solutions
1. Let be a primitive 5-th root of unity in GF (16). Then over GF (16), we can factor x5 1 as
x5 1 = (x 1)(x )(x 2 )(x 3 )(x 4 )
Let be dened as in HW 6 Ex.1. Write all elements of GF (16) in exponential form.
(a) What are th