Coding gain
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). Example. If the uncoded BPSK system in AWGN environment has a bit error rate (BER) of 10 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). Power-limited regime. In the "power-limited regime" (where the nominal spectral efficiency formula_1 [b/2D or b/s/Hz], "i.e." the domain of binary signaling), the effective coding gain formula_2 of a signal set formula_3 at a given target error probability per bit formula_4 is defined as the difference in dB between the formula_5 required to achieve the target formula_4 with formula_3 and the formula_5 required to achieve the target formula_4 with 2-PAM or (2×2)-QAM ("i.e." no coding). The nominal coding gain formula_10 is defined as This definition is normalized so that formula_12  for 2-PAM or (2×2)-QAM. If the average number of nearest neighbors per transmitted bit formula_13 is equal to one, the effective coding gain formula_2 is approximately equal to the nominal coding gain formula_10. However, if formula_16, the effective coding gain formula_2 is less than the nominal coding gain formula_10 by an amount which depends on the steepness of the formula_4 "vs." formula_5 curve at the target formula_4. 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 formula_23 with  parameters formula_24, the nominal spectral efficiency is formula_25 and the nominal coding gain is "kd"/"n". Example. The table below lists the nominal spectral efficiency, nominal coding gain and effective coding gain at formula_26 for Reed–Muller codes of length formula_27: Bandwidth-limited regime. In the "bandwidth-limited regime" (formula_28, "i.e." the domain of non-binary signaling), the effective coding gain formula_2 of a signal set formula_3 at a given target error rate formula_31 is defined as the difference in dB between the formula_32 required to achieve the target formula_31 with formula_3 and the formula_32 required to achieve the target formula_31  with M-PAM or (M×M)-QAM ("i.e." no coding). The nominal coding gain formula_10 is defined as This definition is normalized so that formula_12 for M-PAM or ("M"×"M")-QAM. The UBE becomes where formula_41 is the average number of nearest neighbors per two dimensions. References. MIT OpenCourseWare, 6.451 Principles of Digital Communication II, Lecture Notes sections 5.3, 5.5, 6.3, 6.4