ELEC-E7240 - CODING METHODS
Tutorial 3 on linear codes
0. News
The “Pre-assignment B1” for the lecture is available in mycourses.
1. Problems
1. There is a finite field of order 17. One can represent this field with elements from {0, 1, . . . , 16}
and operate on those elements modulo 17. Use this representation in the following.
(a) How do we know that such a field exists?
(b) Find a primitive element in this field.
(c) Solve the equation 2x + 3 = 2.
(d) Solve the equation x4 = 1.
2. Polynomials.
(a) Express x3 + 1 ∈ GF(2)[x] as a product of binary irreducible polynomials.
(b) Multiply x2 + x + 1 and x2 + x + 2 in the ring GF(3)[x]/(x3 − 1).
3. Consider the binary linear code with the following generator matrix


1 0 1 0 1 1
G =  1 1 0 1 0 1 .
0 1 1 0 0 1
(a)
(b)
(c)
(d)
(e)
(f)
How many codewords does this code have?
What is the minimum distance of the code?
What is the rate of the code?
Encode the message block 1 1 0 using the generator matrix G.
Give a parity check matrix
H for this code. Use H to check whether 1 0 1 0 0 1 is a codeword.
4. Consider the binary code C of all palindromes of length n, that is, words that are the same read
forwards and backwards.
(a) Show that the code C is linear.
(b) For n = 8, describe the codewords (x1 , x2 , . . . , x8 ) using equations. Give a parity check
matrix for this code, and determine the number of errors it detects.
2. Homeworks
H6. Consider an (8, 4) binary linear block code C with codewords (v0 , v1 , v2 , v3 , u0 , u1 , u2 , u3 ), where
{u0 , u1 , u2 , u3 } are the message bits, and {v0 , v1 , v2 , v3 } are the parity-check bits. The parity-bit
equations are as follows: v0 = u0 +u1 +u2 , v1 = u0 +u1 +u3 , v2 = u0 +u2 +u3 , v3 = u1 +u2 +u3 .
(a) Find the generator matrix and the parity-check matrix for this code.
(b) What is the minimum distance of the code? How many errors can it detect? How many
errors can it correct?
H7. Consider polynomials with coefficients in GF(13).
(a) Determine the factorization of the polynomial x6 − 1.
(b) Determine the factorization of the polynomial x7 − 1.
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