Encyclopedia

Coding gain

Wikipedia contributorsAdapted for Fathomly · View authors

2 min read
Source updated February 18, 2025

In coding theory, telecommunications engineering and other 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).

01Example

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 = 4 dB − 2.5 dB = 1.5 dB, due to the code used (in this case BCH).

02Power-limited regime

In the power-limited regime (where the nominal spectral efficiency $\rho \leq 2$ [b/2D or b/s/Hz], i.e. the domain of binary signaling), the effective coding gain $\gamma _{\mathrm {eff} }(A)$ 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

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

This definition is normalized so that $\gamma _{c}(A)=1$ for 2-PAM or (2×2)-QAM. If the average number of nearest neighbors per transmitted bit $K_{b}(A)$ is equal to one, the effective coding gain $\gamma _{\mathrm {eff} }(A)$ is approximately equal to the nominal coding gain $\gamma _{c}(A)$ . However, if $K_{b}(A)>1$ , the effective coding gain $\gamma _{\mathrm {eff} }(A)$ is less than the nominal coding gain $\gamma _{c}(A)$ by an amount which depends on the steepness of the $P_{b}(E)$ vs. $E_{b}/N_{0}$ curve at the target $P_{b}(E)$ . This curve can be plotted using the union bound estimate (UBE)

$P_{b}(E)\approx K_{b}(A)Q\left({\sqrt {\frac {2\gamma _{c}(A)E_{b}}{N_{0}}}}\right),$

where Q is the Gaussian probability-of-error function.

For the special case of a binary linear block code $C$ with parameters $(n,k,d)$ , the nominal spectral efficiency is $\rho =2k/n$ and the nominal coding gain is kd/n.

03Example

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)	$K_{b}$	$\gamma _{\mathrm {eff} }$ (dB)
[8,7,2]	1.75	7/4	2.43	4	2.0
[8,4,4]	1.0	2	3.01	4	2.6
[16,15,2]	1.88	15/8	2.73	8	2.1
[16,11,4]	1.38	11/4	4.39	13	3.7
[16,5,8]	0.63	5/2	3.98	6	3.5
[32,31,2]	1.94	31/16	2.87	16	2.1
[32,26,4]	1.63	13/4	5.12	48	4.0
[32,16,8]	1.00	4	6.02	39	4.9
[32,6,16]	0.37	3	4.77	10	4.2
[64,63,2]	1.97	63/32	2.94	32	1.9
[64,57,4]	1.78	57/16	5.52	183	4.0
[64,42,8]	1.31	21/4	7.20	266	5.6
[64,22,16]	0.69	11/2	7.40	118	6.0
[64,7,32]	0.22	7/2	5.44	18	4.6

04Bandwidth-limited regime

In the bandwidth-limited regime ( $\rho >2~b/2D$ , i.e. the domain of non-binary signaling), the effective coding gain $\gamma _{\mathrm {eff} }(A)$ of a signal set $A$ at a given target error rate $P_{s}(E)$ is defined as the difference in dB between the $SNR_{\mathrm {norm} }$ required to achieve the target $P_{s}(E)$ with $A$ and the $SNR_{\mathrm {norm} }$ 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

$\gamma _{c}(A)={(2^{\rho }-1)d_{\min }^{2}(A) \over 6E_{s}}.$

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

$P_{s}(E)\approx K_{s}(A)Q{\sqrt {3\gamma _{c}(A)SNR_{\mathrm {norm} }}},$

where $K_{s}(A)$ is the average number of nearest neighbors per two dimensions.

Watch videos about Coding gainExplainers and documentaries on YouTube (opens in a new tab)

Sources and credits

This article is adapted from the Wikipedia article “Coding gain”, written by its contributors and licensed under CC BY-SA 4.0. Fathomly has changed the layout, removed citation markers, navigation and maintenance notices, and adjusted punctuation. This adapted version is shared under the same license. For references, see the original article.

Fathomly is not affiliated with or endorsed by the Wikimedia Foundation. Spotted a problem? Tell us.

Coding gain

01Example

02Power-limited regime

03Example

04Bandwidth-limited regime

Sources and credits

Related topics

Channel capacity

Eb/N0