Exact BER computation for the cross 32-QAM constellation

When the number of bits per symbol is odd, the peak and average power of transmission can be reduced by using cross quadrature amplitude modulations (QAMs) instead of rectangular QAMs. Since perfect Gray coding is not possible for cross QAMs, using Smith-style Gray coding, we derive in this paper the exact bit error rate (BER) for the cross 32-QAM constellation over additive white Gaussian noise (AWGN) and Rayleigh fading channels.