2007 - Exercises II.
1. Decide which of the following codes is linear. Find the standard form of generator
matrix for codes that are linear.
(a) quaternary code Cx = {000,123, 202, 321}
(b) 6-ary code C2 = {00, 03, 43, 23, 20, 40}
(c) ternary code C3 = {000, 221,112}
2. Compare the set of perfect codes with the set of maximum distance separable
codes (ie. use C , = , . . . ) .
3. Let C be a binary code with generator matrix
G =
1 0 0 0
0 1 1 1
(a) Write a generator matrix for the code obtained from C by puncturing the
1st and 4th coordinate.
(b) Write a generator matrix for the extended code C.
(If C is a linear code over FTM then C = {x\X2 ˇ ˇ ˇ xnxn+\ E FTM+1
|xi... xn E
C and X\ + ... xn + xn+\ = 0} is the extended code.)
(c) Describe the code that is obtained from a linear code first by extending
this code and then puncturing on the new coordinate. What happens if we
change the order of these operations?
4. Let C be a binary linear [4, 2]-code such that C = C^. Show that C contains
at least two words of weight 2.
5. Let C be a perfect binary linear [7, 4]-code. Find the value of d. Suppose that C
is transmitted over a binary symmetric channel with error probability p = 0.01
using the syndrome decoding strategy. Calculate the word error probability.
6. Consider the linear code C over the field Z7 spanned by the codewords 43352,
24545 and 31433.
(a) Find a generator matrix in the standard form for C.
(b) How many different messages can C encode?
(c) Show that a received word containing at most one error can be corrected.