Error-Control Coding – Exercise –

Error-Control Coding – Exercise –
Hochschule Wismar, University of Technology, Business and Design, Prof. Dr.-Ing. habil. A. Ahrens
Baltic Summer School 2011, Riga, Latvia, 12.08. – 27.08.2011
Exercise 1
Hadamard-code
a) Hadamard-codes belong to the class of linear block codes. The field of code words C
of a Hadamard-code can be generated in matlab with the command hadamard(M)
with M possible code words. The code words have the length n = M. Note, that the
result doesn’t consist of zeroes and ones, but of the symbols ’+1’ and ’-1’. Generate
a Hadamard-code for M = 8 and determine the minimum Hamming distance.
b) For coding a group of m = 3 information bits is mapped on one out of M = 8
code words. The 3-bit-words are for the present converted into decimal numbers
that are applied for the addressing by the matlab-command bi2de (requires the
communications toolbox). Alternatively, decimal numbers in the range of 1 up to M
can be chosen by random and converted into 3-bit-words with the command de2bi.
Determine the valid codewords!.
Exercise 2
Error correction
Given is a (n, m) block code with the minimal distance h min = 8.
a) Determine the maximal number of correctable errors and the number of detectable
errors at pure error detection.
b) Demonstrate by illustration in the field of code words, how many possibilities of
variation of a code word have to be taken into consideration at the transmission
over a disturbed channel.
Exercise 3
Generator and parity check matrices
State the generator matrix as well as the parity check matrix for a (n, 1, n) repetition
code with n = 4.

Exercise 4
Modification of linear block codes
A (5, 4) linear block code is constructed by adapting a (7, 4) block code. The (7, 4) code
can be described by the following generator matrix
⎛
⎞
0 1 0 1 1 0 0
G = ⎜ 1 0 1 0 1 0 0
⎟
⎝ 0 1 1 0 0 1 0 ⎠ .
1 1 0 0 0 0 1
a) Construct the codewords of the (5, 4) code and list them!
b) What is the minimum distance of the (5, 4) code!
Exercise 5
Syndrome decoding
a) State the number of syndromes of the (7, 4, 3)-Hamming code and compare it with
the number of correctable error patterns.
b) The word d = (1 1 0 1 0 0 1) is found at the receiver. Which information word i was
sent with the greatest probability?
Exercise 6
Coding program
Write a matlab-programm which codes and again decodes a certain number of input
data bits. Besides it shall be possible to insert errors before the decoding. The (5, 2, 3)
code shall be used.
Hint: The (5, 2, 3) code maps m = 2 information symbols onto n = 5 code symbols.
Within the matlab-program, m information symbols are randomly chosen using randint
and encoded by the generator matrix. A randomly determined error vector is added and
the syndrome is calculated. Please notice: all calculations have to be executed within
GF (2).