How does the CRC aid improve SCL performance?

Without CRC, the SCL decoder outputs the path with the highest metric, which is not always correct. With a CRC appended to the information bits before encoding, the decoder checks all L surviving paths against the CRC and picks the one that passes. If multiple paths pass (rare), it picks the one with the highest metric. If none pass, it declares an error. This CRC-aided selection effectively gives near-ML (Maximum Likelihood) performance for list sizes as small as L=8 to L=32. The 5G NR specification uses 6-bit, 11-bit, or 24-bit CRC depending on the channel and payload size.

How does CA-SCL compare to LDPC in 5G NR?

5G NR uses polar codes with CA-SCL for control channels (PDCCH, PUCCH) and LDPC for data channels (PDSCH, PUSCH). Polar codes with CA-SCL outperform LDPC at short block lengths (under 200 bits), which is typical for control information. LDPC outperforms polar codes at long block lengths (1000+ bits), typical for user data. The crossover point is around 200-500 bits depending on the code rate. CA-SCL with L=8 and 24-bit CRC achieves within 0.1 dB of finite-length capacity bounds at control channel block lengths.

Signal Processing

CA-SCL

Q: Why does polar code decoding need a list?

Arikan's original Successive Cancellation (SC) decoder makes hard decisions bit by bit, left to right. If it makes a wrong decision on an early bit, all subsequent bits are corrupted because polar decoding is serial and each bit depends on the previous decisions. SCL keeps L parallel copies (paths) of the decoding process. At each bit position where the decision is uncertain, it forks into both possibilities, keeping the L most likely paths based on path metrics. With L=8, the decoder almost always has the correct path in its list. The CRC then identifies which path is correct.

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CRC-Aided Successive Cancellation List decoding: the standard algorithm for decoding polar codes in 5G NR control channels (PDCCH, PUCCH). CA-SCL maintains L parallel decoding paths through the polar code factor graph, branching at each uncertain bit decision, and uses a CRC appended to the information bits to select the correct path from the surviving list. With list sizes of L = 8 to 32, CA-SCL achieves performance within 0.1 dB of finite-length capacity bounds at short block lengths.

Category: Signal Processing

Full name: CRC-Aided Successive Cancellation List

5G NR usage: PDCCH, PUCCH (control)

Understanding CA-SCL Decoding

Polar codes, invented by Erdal Arikan in 2009, are the first provably capacity-achieving codes with explicit construction. They work by "polarizing" a set of identical binary channels into channels that are either nearly perfect (used for information bits) or nearly useless (frozen to known values). The original SC decoder decodes bits sequentially, but a single wrong decision propagates through the entire codeword.

SCL decoding fixes this by maintaining L candidate paths. At each bit position, the decoder computes the likelihood of both 0 and 1 decisions. If the decision is uncertain (the likelihoods are close), it forks the path into two, keeping the L paths with the highest cumulative path metrics. After all N bits are decoded, L candidate codewords remain. Without CRC, the best-metric path is chosen (often wrong). With CRC, the decoder checks all L paths and selects the one passing the CRC check, dramatically improving accuracy.

CA-SCL Algorithm

Path Metric Update:
PM_l(i) = PM_l(i−1) + ln(1 + e^{−(1−2û_i)×LLR_i})
where û_i is the bit decision on path l

List Pruning:
At each fork: 2L candidates → keep L with lowest PM

CRC Selection:
After all bits decoded, check each path's CRC
Output = path with valid CRC and lowest PM

Complexity:
O(L × N × log N), where N = code length

Example: N=512, K=56, L=8, CRC-24 → BLER = 10⁻³ at Eb/No ≈ 1.5 dB (within 0.1 dB of capacity).

5G NR Channel Coding Comparison

Code	Decoder	5G NR Channel	Block Length	Latency
Polar + CA-SCL	CA-SCL (L=8)	PDCCH, PUCCH	12-1024 bits	Low (μs)
LDPC	BP / Min-Sum	PDSCH, PUSCH	100-8448 bits	Moderate
Reed-Muller	ML / correlation	PUCCH (tiny payload)	2-11 bits	Very low
Turbo (4G LTE)	MAP / Max-Log-MAP	LTE PDSCH/PUSCH	40-6144 bits	High

Common Questions

Frequently Asked Questions

Why does polar code decoding need a list?

SC makes hard decisions bit-by-bit. A wrong early decision corrupts all subsequent bits. SCL keeps L parallel paths, forking at uncertain positions. With L=8, the correct path is almost always in the list. The CRC then identifies which one is right.

How does CRC aid improve SCL performance?

Without CRC, SCL picks the highest-metric path (not always correct). With CRC, all L survivors are checked and the one passing CRC is selected. This gives near-ML performance for L=8 to 32. 5G NR uses 6-bit, 11-bit, or 24-bit CRC depending on channel and payload size.

How does CA-SCL compare to LDPC in 5G?

Polar+CA-SCL wins at short blocks (<200 bits, control info). LDPC wins at long blocks (1000+ bits, user data). Crossover is 200-500 bits. This is why 5G uses both: polar for control, LDPC for data.

Related Terms

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