Signal Processing

CRC-Aided SC List

Q: How does the CRC actually help an SC list decoder choose the right path?

Plain successive-cancellation list (SCL) decoding finishes with L surviving candidate paths and normally outputs the one with the best path metric, but that path is sometimes not the transmitted codeword even when a correct candidate exists in the list. CRC-aided SCL appends a short CRC (typically 6, 11, 16, or 24 bits in 5G NR) to the message before polar encoding. After SCL produces its L candidates, the decoder recomputes the CRC on each candidate and discards every path that fails the check. Among the CRC-passing paths it then selects the one with the best path metric. The CRC effectively acts as a genie-aided selector: as long as the true codeword is among the L survivors and no wrong path also passes the CRC, the decoder picks correctly. This is why CA-SCL closes most of the gap to maximum-likelihood decoding that ordinary SCL leaves open.

Q: What list size L and CRC length does 5G NR use for polar codes?

The 3GPP NR specification (TS 38.212) uses an 11-bit CRC for the uplink control channel (PUCCH/PUSCH UCI) and the downlink PDCCH carries a 24-bit CRC, while the broadcast PBCH also uses a 24-bit CRC. Commercial receiver implementations typically run CA-SCL with a list size of L = 8, which gives a strong block error rate while keeping decoder latency and area manageable. Increasing L to 16 or 32 yields diminishing returns of a few tenths of a dB because the limiting factor becomes whether the correct path survives the list rather than CRC selection. The PDCCH also uses distributed CRC bit interleaving so the decoder can perform early termination, aborting paths that already violate the partial CRC before the full block is decoded.

Q: Why can a longer CRC sometimes hurt CA-SCL performance?

There are two competing effects. A longer CRC reduces the false-alarm probability that a wrong candidate path accidentally passes the check, which is roughly 2 to the power of minus the CRC length per surviving path, so a 24-bit CRC almost never lets an incorrect path slip through. However, those CRC bits are pure overhead that steal coding gain from the polar code itself: for a fixed transmitted block length, a longer CRC means fewer redundancy bits available for the actual error correction, raising the effective code rate. At short block lengths the overhead penalty dominates, so a 6 to 11 bit CRC is often optimal, while at larger blocks or when very low undetected-error rates are required a longer CRC is justified. The design choice balances false-alarm rate against rate loss.

Q: How does CA-SCL latency and complexity compare to LDPC decoding?

CA-SCL complexity scales as O(L times N log N) where N is the block length and L is the list size, and it is inherently sequential because successive cancellation decodes bits one after another along the polar tree. That serial nature gives polar CA-SCL higher latency per bit than a fully parallel LDPC layered belief-propagation decoder, which is why 5G NR assigns polar codes to short control blocks (up to roughly 1706 bits payload) and LDPC to the high-throughput data channel. For the small control payloads, CA-SCL with L = 8 delivers excellent block error rate at very low SNR with acceptable latency, whereas LDPC excels at the large data blocks where its parallelism and throughput dominate.

/C-R-C ay-did S-C list/

Often abbreviated CA-SCL, this is the practical decoding algorithm that made polar codes competitive enough to be standardized for 5G NR control channels. A successive-cancellation list decoder retains the L most likely candidate codeword paths instead of committing to a single hard decision, and a short cyclic redundancy check concatenated with the message is then used to pick the one surviving path whose CRC verifies. Because the CRC bits also raise the minimum distance among the surviving candidates, CA-SCL with a list size of L = 8 typically reduces block error rate by one to two orders of magnitude over plain SCL at the same signal-to-noise ratio, closing most of the gap to maximum-likelihood forward error correction.

Category: Signal Processing

Typical list size L: 8 (up to 32)

5G NR CRC: 11 or 24 bits

How CRC Concatenation Rescues List Decoding

Arikan's original successive-cancellation (SC) decoder for polar codes is elegant and low-complexity but performs poorly at finite block lengths because a single early bit error propagates through the rest of the decoding tree. List decoding fixes the propagation problem by exploring multiple hypotheses: at each information bit the decoder forks both the 0 and 1 hypotheses, keeps the L paths with the best accumulated metric, and prunes the rest. The flaw is in the final step. After SCL produces its L survivors, the path with the best metric is not always the transmitted codeword, even when a correct candidate sits somewhere in the list. Researchers observed that the true codeword was frequently present but simply not selected.

CRC-aided SC list decoding solves the selection problem by concatenating a short cyclic redundancy check with the message before polar encoding. The CRC turns into an error-detecting outer code. After SCL finishes, the decoder recomputes the CRC on each of the L candidates and immediately discards any path that fails, then chooses the best-metric path among those that pass. As long as the correct path survives in the list and no incorrect path happens to satisfy the CRC, the decoder selects the right codeword. The probability that a wrong path falsely passes is approximately 2^−r per candidate for an r-bit CRC, so even a modest CRC makes false selection rare.

The second, less obvious benefit is distance: the CRC and polar code together behave like a concatenated code whose minimum distance exceeds that of the polar code alone. This is why CA-SCL beats not only ordinary SCL but, at practical list sizes, even approaches the maximum-likelihood performance bound of the standalone polar code. The combined gain is what convinced 3GPP to adopt polar codes with CA-SCL for the 5G NR uplink and downlink control channels.

Path Metric and CRC Selection

Log-likelihood path metric (lower is better):
PM_l⁽ⁱ⁾ = PM_l⁽ⁱ⁻¹⁾ + ln(1 + e^{−(1−2û_i)·L_i})

Hardware approximation:
PM ≈ PM_prev + |L_i| if û_i ≠ ½(1 − sign L_i), else unchanged

Final CRC-aided selection:
ŵ = arg min_l { PM_l^(N) : CRC(message bits of path l) = 0 }

False-pass probability of a wrong path:
P_fa ≈ 2^−r per surviving candidate, r = CRC length in bits

Where L_i is the channel log-likelihood ratio at bit i, û_i is the trial hard decision, N is the polar block length, and L is the list size. Example: r = 11 bit CRC, L = 8 survivors → P_fa ≈ 8 × 2⁻¹¹ ≈ 0.004.

Decoder Comparison at Short Block Lengths

Decoder	Complexity	BLER vs. SC (at 1e−3)	Latency	5G NR role	Notes
SC (plain)	O(N log N)	baseline (worst)	Low, sequential	Not used directly	Error propagation hurts
SCL, L=8	O(L·N log N)	~1 to 2 dB better	Moderate	Intermediate step	Best-metric selection fails sometimes
CA-SCL, L=8	O(L·N log N)	~2.5 to 3.5 dB better	Moderate	PDCCH, PUCCH, PBCH	CRC picks the correct survivor
CA-SCL, L=32	O(L·N log N)	~0.2 to 0.4 dB over L=8	Higher	Lab / reference	Diminishing returns
LDPC (BP)	O(N·iter), parallel	n/a (data channel)	Low, parallel	PDSCH / PUSCH data	Better for long blocks

Common Questions

Frequently Asked Questions

How does the CRC actually help an SC list decoder choose the right path?

Plain SCL finishes with L survivors and normally outputs the best path-metric candidate, but that path is sometimes wrong even when a correct candidate exists in the list. CA-SCL appends a short CRC (6, 11, 16, or 24 bits in 5G NR) to the message before encoding. After SCL produces its L candidates, the decoder recomputes the CRC on each and discards every failing path, then picks the best-metric path among the CRC-passing ones. As long as the true codeword survives and no wrong path also passes the CRC, selection is correct, which closes most of the gap to maximum-likelihood decoding.

What list size L and CRC length does 5G NR use for polar codes?

3GPP TS 38.212 uses an 11-bit CRC for uplink control (PUCCH/PUSCH UCI) and a 24-bit CRC for the downlink PDCCH and the PBCH. Commercial receivers typically run CA-SCL with L = 8, balancing block error rate against latency and area. Pushing L to 16 or 32 yields only a few tenths of a dB because survival of the correct path, not CRC selection, becomes the limiter. PDCCH also uses distributed CRC interleaving so the decoder can early-terminate paths that violate the partial CRC.

Why can a longer CRC sometimes hurt CA-SCL performance?

Two effects compete. A longer CRC lowers the false-pass probability of a wrong path, roughly 2^−r per survivor, so a 24-bit CRC almost never lets a bad path through. But those CRC bits are overhead that steal coding gain: for a fixed transmitted block, more CRC bits leave fewer redundancy bits for the polar code, raising the effective rate. At short blocks the overhead penalty dominates, so a 6 to 11 bit CRC is often optimal; longer CRCs are justified only for larger blocks or very low undetected-error targets.

How does CA-SCL latency and complexity compare to LDPC decoding?

CA-SCL scales as O(L·N log N) and is inherently sequential because successive cancellation decodes bits one at a time along the polar tree, giving higher latency per bit than a fully parallel LDPC belief-propagation decoder. That is why 5G NR uses polar CA-SCL for short control blocks (up to about 1706 payload bits) and LDPC for the high-throughput data channel. For small control payloads, CA-SCL with L = 8 delivers excellent block error rate at low SNR with acceptable latency, while LDPC dominates on the large data blocks where its parallelism wins.

Related Terms

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