Coding Gain - Power-limited Regime

In the power-limited regime (where the nominal spectral efficiency, i.e. the domain of binary signaling), the effective coding gain of a signal set at a given target error probability per bit is defined as the difference in dB between the required to achieve the target with and the required to achieve the target with 2-PAM or (2×2)-QAM (i.e. no coding). The nominal coding gain is defined as

This definition is normalized so that for 2-PAM or (2×2)-QAM. If the average number of nearest neighbors per transmitted bit is equal to one, the effective coding gain is approximately equal to the nominal coding gain . However, if, the effective coding gain is less than the nominal coding gain by an amount which depends on the steepness of the vs. curve at the target . This curve can be plotted using the union bound estimate (UBE)

where denotes the Gaussian probability of error function.

For the special case of a binary linear block code with parameters, the nominal spectral efficiency is and the nominal coding gain is kd/n.