Consider a QAM system designed to transmit over a bandwidth of $3 \ \mathrm{kHz}$. The channel's power constraint imposes a maximum $\mathrm{SNR}$ of $30 \ \mathrm{dB}$. The system can tolerate a probability of error of $10^{−6}$. I want to determine the maximum throughput of the system in bits per second.
Let's call the throughput $T$. If $M$ is the constellation size and $"$ is the bandwidth, then I know that $T=MW$. I've calculated the value of $M$ using theformula $$M = \log_2 \left( 1 - \frac{1.5\mathrm{SNR}}{\ln(p_e)}\right)$$ Then I approximated $M$ to the nearest power of $2$, which is $8$. Therefore, $T = 24000 \ \mathrm{\frac{bits}{sec}}$ but it is a wrong answer. Can anyone tell me what I have done wrong?