Coding gain is the reduction in required Eb/N0 to achieve a target BER, compared to uncoded transmission. For BPSK at BER = 1e-6, uncoded Eb/N0 = 10.5 dB. With rate-1/2 LDPC: Eb/N0 approximately 2 dB. Coding gain = 10.5 - 2 = 8.5 dB. This means the system can operate with 8.5 dB less signal power (or equivalently, at 2.7x longer range for the same power). The tradeoff: coding adds overhead. Rate-1/2 code doubles the required bandwidth. The net coding gain (accounting for bandwidth expansion) is: net gain = coding gain - 10*log10(1/R). For R=1/2: net gain = 8.5 - 3 = 5.5 dB. This is still a massive improvement, which is why every modern wireless system uses FEC.

What FEC codes are used in 5G NR?

5G NR uses two FEC code families: LDPC (Low-Density Parity-Check) for data channels (PDSCH/PUSCH): based on quasi-cyclic base graphs (BG1 for large blocks, BG2 for small blocks). Code rates from 1/5 to 948/1024. Information block sizes from 40 to 8448 bits. Achieves within 0.5 dB of Shannon limit at long block lengths. Iterative belief propagation decoding. Polar codes for control channels (PDCCH/PUCCH/PBCH): capacity-achieving on BMS channels. Uses successive cancellation list (SCL) decoding with CRC-aided selection. Block sizes from 12 to 1706 bits. Better than LDPC at short block lengths, which is why they are used for control information. 5G NR was the first commercial system to deploy polar codes.

How close are modern codes to Shannon limit?

Shannon limit (capacity): Eb/N0 > -1.59 dB for reliable communication at any rate. At rate 1/2 BPSK: Shannon limit = approximately 0.19 dB Eb/N0. Modern achievements: LDPC (5G NR, Wi-Fi 6): approximately 0.5 dB from Shannon at long block lengths. Polar codes: provably capacity-achieving (asymptotically), approximately 1 dB gap at practical block lengths. Turbo codes (LTE): approximately 1 dB from Shannon. Historical progress: 1993 turbo codes: 0.7 dB gap at BER=1e-5 (revolutionary at the time). 2001 LDPC rediscovery: matched turbo with simpler hardware. 2009 polar codes: first provably capacity-achieving construction. 2018: 5G NR deploys both LDPC and polar in the first commercial system to approach Shannon limit.

Channel Coding

Forward Error Correction (FEC)

/for-werd er-or kor-ek-shun/

FEC adds redundancy for error correction without retransmission. Code rate R = k/n. Coding gain: 8-10 dB for LDPC at BER=10^-6. 5G NR: LDPC for data, polar codes for control. Turbo codes in LTE. Modern LDPC operates within 0.5 dB of Shannon limit. Every wireless system uses FEC; it is the single most impactful technology for reliable digital communication.

Category: Channel Coding

LDPC gain: 8-10 dB

Shannon gap: 0.5 dB

Understanding FEC

FEC is the reason modern wireless systems work at all. Without coding, a 256-QAM signal would need 29 dB of SNR for BER=10^-6. With LDPC coding, it needs about 19 dB. That 10 dB coding gain means either 3x more range, 10x less transmit power, or 10x more capacity at the same power. Every wireless standard since GSM has relied on FEC as the cornerstone of reliable communication. The evolution from convolutional codes to turbo codes to LDPC and polar codes represents one of the greatest achievements in information theory and engineering.

FEC Fundamentals

Shannon limit:
C = B·log₂(1+SNR)

Coding gain:
G_c = (Eb/N0)_uncoded − (Eb/N0)_coded dB

Code rate:
R = k/n (information bits / total bits)

Net data rate:
R_net = R_gross × R_code

FEC Code Comparison

Code	Coding Gain	Shannon Gap	Decoder	Standard
Convolutional	5-7 dB	3-5 dB	Viterbi	GSM, 802.11a/g
Reed-Solomon	4-6 dB	4-6 dB	Algebraic	DVB-S, fiber
Turbo	7-9 dB	~1 dB	Iterative MAP	LTE, 3G
LDPC	8-10 dB	~0.5 dB	Belief propagation	5G NR, Wi-Fi 6
Polar	7-9 dB	~1 dB	SCL + CRC	5G NR control

Key Equations

Decibel conversion:
Power: dB = 10log(P₂/P₁)
Voltage: dB = 20log(V₂/V₁)

dBm to watts:
P(W) = 10^{(dBm−30)/10}
0 dBm = 1 mW, +30 dBm = 1 W

Wavelength:
λ = c/f = 300/f(MHz) meters

Comparison

Code	Rate	Gain	Latency	Standard
Conv 1/2	0.5	5–7 dB	Low	Legacy WiFi
Turbo	1/3–4/5	7–9 dB	Medium	3G/4G
LDPC	1/2–5/6	8–10 dB	Medium	WiFi 6, 5G, DVB
Polar	1/2–5/6	8–10 dB	Low	5G control
RS(255,239)	0.937	3–5 dB	Low	Optical, storage

Common Questions

Frequently Asked Questions

Coding gain?

Eb/N0 reduction vs. uncoded. LDPC R=1/2: ~8.5 dB at BER=1e-6. Means 3x more range or 10x less power. Net coding gain accounts for BW expansion: NCG = gain - 10*log(1/R).

5G NR codes?

LDPC for data (PDSCH/PUSCH): rates 1/5 to 948/1024, within 0.5 dB of Shannon. Polar for control (PDCCH/PUCCH/PBCH): capacity-achieving, better than LDPC at short blocks. First commercial polar code deployment.

Shannon limit?

Minimum Eb/N0 for error-free communication: -1.59 dB. LDPC: 0.5 dB gap. Turbo: ~1 dB. Convolutional: 3-5 dB. 30 years of coding research closed the gap from 5 dB to 0.5 dB.

Related Terms

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