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In [[Coding theory|coding theory]], [[Engineering:Telecommunications engineering|telecommunications engineering]] and other related engineering problems, '''coding gain''' is the measure in the difference between the [[Physics:Signal-to-noise ratio|signal-to-noise ratio]] (SNR) levels between the uncoded system and coded system required to reach the same [[Bit error rate|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|bit error rate]] (BER) of 10<sup>−2</sup> at the SNR level 4&nbsp;[[Physics:Decibel|dB]], and the corresponding coded (e.g., [[BCH code|BCH]]) system has the same BER at an SNR of 2.5&nbsp;dB, then we say the ''coding gain'' = {{nowrap|1=4 dB − 2.5 dB = 1.5 dB}}, due to the code used (in this case BCH).

==Power-limited regime==
In the ''power-limited regime'' (where the nominal [[Spectral efficiency|spectral efficiency]] <math>\rho \le 2</math> [b/2D or b/s/Hz], ''i.e.'' the domain of binary signaling), the effective coding gain <math>\gamma_\mathrm{eff}(A)</math> of a signal set <math>A</math> at a given target error probability per bit <math>P_b(E)</math> is defined as the difference in dB between the <math>E_b/N_0</math> required to achieve the target <math>P_b(E)</math> with <math>A</math> and the <math>E_b/N_0</math> required to achieve the target <math>P_b(E)</math> with 2-[[Engineering:Pulse-amplitude modulation|PAM]] or (2&times;2)-[[Quadrature amplitude modulation|QAM]] (''i.e.'' no coding). The nominal coding gain <math>\gamma_c(A)</math> is defined as

: <math>\gamma_c(A) = \frac{d^2_{\min}(A)}{4E_b}.</math>

This definition is normalized so that <math>\gamma_c(A) = 1</math> for 2-PAM or (2&times;2)-QAM. If the average number of nearest neighbors per transmitted bit <math>K_b(A)</math> is equal to one, the effective coding gain <math>\gamma_\mathrm{eff}(A)</math> is approximately equal to the nominal coding gain <math>\gamma_c(A)</math>. However, if <math>K_b(A)>1</math>, the effective coding gain <math>\gamma_\mathrm{eff}(A)</math> is less than the nominal coding gain <math>\gamma_c(A)</math> by an amount which depends on the steepness of the <math>P_b(E)</math> ''vs.'' <math>E_b/N_0</math> curve at the target <math>P_b(E)</math>. This curve can be plotted using the union bound estimate (UBE)

: <math>P_b(E) \approx K_b(A)Q\left(\sqrt{\frac{2\gamma_c(A)E_b}{N_0}}\right),</math>

where ''Q'' is the [[Error function|Gaussian probability-of-error function]].

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

==Example==
The table below lists the nominal spectral efficiency, nominal coding gain and effective coding gain at <math>P_b(E) \approx 10^{-5}</math> for [[Reed–Muller code]]s of length <math>n \le 64</math>:

{| class="wikitable"
 ! Code !! <math>\rho</math> !! <math>\gamma_c</math> !! <math>\gamma_c</math> (dB) !! <math>K_b</math> !! <math>\gamma_\mathrm{eff}</math> (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
 |-
 |}

==Bandwidth-limited regime==
In the ''bandwidth-limited regime'' (<math>\rho > 2~b/2D</math>, ''i.e.'' the domain of non-binary signaling), the effective coding gain <math>\gamma_\mathrm{eff}(A)</math> of a signal set <math>A</math> at a given target error rate <math>P_s(E)</math> is defined as the difference in dB between the <math>SNR_\mathrm{norm}</math> required to achieve the target <math>P_s(E)</math> with <math>A</math> and the <math>SNR_\mathrm{norm}</math> required to achieve the target <math>P_s(E)</math> with M-[[Engineering:Pulse-amplitude modulation|PAM]] or (M&times;M)-[[Quadrature amplitude modulation|QAM]] (''i.e.'' no coding). The nominal coding gain <math>\gamma_c(A)</math> is defined as

: <math>\gamma_c(A) = {(2^\rho - 1)d^2_{\min} (A) \over 6E_s}.</math>

This definition is normalized so that <math>\gamma_c(A) = 1</math> for M-PAM or (''M''&times;''M'')-QAM. The UBE becomes

: <math>P_s(E) \approx K_s(A)Q\sqrt{3\gamma_c(A)SNR_\mathrm{norm}},</math>

where <math>K_s(A)</math> is the average number of nearest neighbors per two dimensions.

==See also==
*[[Engineering:Channel capacity|Channel capacity]]
*Eb/N0

==References==
[http://ocw.mit.edu MIT OpenCourseWare], 6.451 Principles of Digital Communication II, Lecture Notes sections 5.3, 5.5, 6.3, 6.4

[[Category:Coding theory]]
[[Category:Error detection and correction]]

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