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Figure 9 Hierarchical minimum distance performance for 8-PSK and 8-ary example codebook (zero mean).

Figure 10 Hierarchical minimum distance performance for 16-QAM and 16-ary example codebook (zero mean).

It is obvious from this figure that the increasing value of the mean of the alphabet degrades the minimum distance performance.

Our objective becomes to determine the matrix Σ subject to the power constraint (12) in order to improve the minimum distance performance.

Note that the distance shortening at is generally inevitable [6]. Figure 8 Hierarchical minimum distance performance for QPSK and 4-ary example codebook (zero-mean).

The MAC stage channel parametrization of the 2-WRC system with HDF strategy affects the decision regions at the relay as well as the overall system performance (which is influenced by the minimum distance performance of the chosen codebooks).

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Later several works, on several channels, have been carried out in order to afford very large minimum distance and exhibit performance approaching Shannon's limit.

This is the reason that the performance improves as the minimum distance of the code increases.

Here, we use the method in[25] which can achieve better performance by maximizing the minimum distance between the codewords.

Also, since the performance of the designed codes is characterized by the minimum distance, it is not necessary to compare it with the performance of other codes.

Figure 1 shows the PLR performance for three codes with the same minimum distance of three and the same packet size of bits but different code rates.

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