Information theory and error control coding/Teoria da informação e códigos corretores
de erros
CETUC/PUC-Rio - Prof. Rodrigo de Lamare
Tutorial Questions/Lista de Exercícios - 6
1. Consider a BCH code with code length n=15 and generator polynomial g(X) =
(1+X)(1+X+X4)(1+X+X2+X3+X4) that operates over the GF(24) described below.
a) What is the minimum Hamming distance of the code, its error correction capability
and the rate of the code? Explain.
b) Consider the message vector m = [1 0 0 0 0 0] . Encode it with the generator
polynomial given above.
c) For the received vector r = [0 1 0 0 1 1 1 0 1 1 0 0 0 0 0] compute the syndrome
values and the syndrome polynomial.
d) Perform the decoding of the received polynomial r(X) = X+ X4 + X5 + X6 + X8 + X9.
2. Consider a Reed-Solomon code with code length n=7 and message length k = 3 that
operates over the GF(23) described below.
a) Consider the received vector r = (000 000 100 000 100 000 000) and write down the
received polynomial r(x) and the syndrome
b) Compute the error location polynomial σ(x) with the Berlekamp-Massey algorithm.
c) Determine the error positions and the error values.
3. Develop a Matlab code to simulate BCH and Reed-Solomon codes with arbitrary
parameters (n,k) using BPSK modulation and additive white Gaussian noise. Suggestion: use
Matlab´s communication toolbox.
a) Plot the bit error ratio (BER) against the signal-to-noise ratio (SNR) for a range of
values. Suggestion: pick 5 or 6 SNR points, use 2 different (n,k) for each coding scheme.
b) Compare the BER x SNR performance of different codes with that of uncoded
systems.