In coding theory and related engineering problems, coding gain is the measure in the difference between the signal-to-noise ratio (SNR) levels between the uncoded system and coded system required to reach the same bit error rate (BER) levels when used with the error correcting code (ECC).
If the uncoded BPSK system in AWGN environment has a bit error rate (BER) of 10−2 at the SNR level 4 dB, and the corresponding coded (e.g., BCH) system has the same BER at an SNR of 2.5 dB, then we say the coding gain = , due to the code used (in this case BCH).
In the power-limited regime (where the nominal spectral efficiency [b/2D or b/s/Hz], 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 Q is 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.
The table below lists the nominal spectral efficiency, nominal coding gain and effective coding gain at for Reed–Muller codes of length :
In the bandwidth-limited regime (, i.e. the domain of non-binary signaling), the effective coding gain of a signal set at a given target error rate is defined as the difference in dB between the required to achieve the target with and the required to achieve the target with M-PAM or (M×M)-QAM (i.e. no coding). The nominal coding gain is defined as
This definition is normalized so that for M-PAM or (M×M)-QAM. The UBE becomes
where is the average number of nearest neighbors per two dimensions.