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 $10^{-3}$ at the SNR level 3dB, and the corresponding coded (e.g., BCH) system has the same BER at an SNR level of 1.5dB, then we say the coding gain = 3dB-1.5dB = 1.5dB, due to the code used (in this case BCH).

Power-limited regime

In the power-limited regime ( $\rho \leq 2b/2D$ ), the effective coding gain $\gamma _{eff}(A)$ of a signal set $A$ at a given target error probability per bit $P_{b}(E)$ is defined as the difference in dB between the $E_{b}/N_{0}$ required to achieve the target $P_{b}(E)$ with $A$ and the $E_{b}/N_{0}$ required to achieve the target $P_{b}(E)$ with 2-PAM or (2×2)-QAM (i.e. no coding). The nominal coding gain $\gamma _{c}(A)$ is defined as

\gamma _{c}(A)={d_{\min(}^{2}A) \over 4E_{b}}.

This definition is normalized so that $\gamma _{c}(A)=1$ for 2-PAM or (2×2)-QAM.

For the special case of a binary linear block code $C$ with parameters $[n,k,d]$ , the nominal spectral efficiency is $\rho =2k/n$ (b/2D or b/s/Hz) 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 $P_{b}(E)\approx 10^{-5}$ for Reed-Muller codes of length $n\leq 64$ :

Code	$\rho$	$\gamma _{c}$	$\gamma _{c}$ (dB)	$\gamma _{eff}$ (dB)
[8,7,2]	1.75	7/4	2.43	2.0
[8,4,4]	1.0	2	3.01	2.6
[16,15,2]	1.88	15/8	2.73	2.1
[16,11,4]	1.38	11/4	4.39	3.7
[16,5,8]	0.63	5/2	3.98	3.5
[32,31,2]	1.94	31/16	2.87	2.1
[32,26,4]	1.63	13/4	5.12	4.0
[32,16,8]	1.00	4	6.02	4.9
[32,6,16]	0.37	3	4.77	4.2
[64,63,2]	1.97	63/32	2.94	1.9
[64,57,4]	1.78	57/16	5.52	4.0
[64,42,8]	1.31	21/4	7.20	5.6
[64,22,16]	0.69	11/2	7.40	6.0
[64,7,32]	0.22	7/2	5.44	4.6

Bandwidth-limited regime

In the bandwidth-limited regime ( $\rho >2b/2D$ ), the effective coding gain $\gamma _{eff}(A)$ of a signal set $A$ at a given target error rate $P_{s}(E)$ is defined as the difference in dB between the $SNR_{norm}$ required to achieve the target $P_{s}(E)$ with $A$ and the $SNR_{norm}$ required to achieve the target $P_{s}(E)$ with M-PAM or (M×M)-QAM (i.e. no coding). The nominal coding gain $\gamma _{c}(A)$ is defined as