Improving the decoding performance of CA-polar codes

We investigate the use of modern code-agnostic decoders to convert CA-SCL from an incomplete decoder to a complete one. When CA-SCL fails to identify a codeword that passes the CRC check, we apply a code-agnostic decoder that identifies a codeword that satisfies the CRC. We establish that this approach gives gains of up to 0.2 dB in block error rate for CA-Polar codes from the 5G New Radio standard. If, instead, the message had been encoded in a systematic CA-polar code, the gain improves to 0.2 ~ 1dB. Leveraging recent developments in blockwise soft output, we additionally establish that it is possible to control the undetected error rate even when using the CRC for error correction.

💡 Research Summary

The paper addresses a fundamental limitation of CRC‑aided polar (CA‑Polar) codes as used in the 5G New Radio standard: the conventional CRC‑Aided Successive Cancellation List (CA‑SCL) decoder is an incomplete decoder that declares a failure whenever none of the list candidates pass the CRC check. This behavior is undesirable for ultra‑reliable low‑latency communication (URLLC) because it can leave the receiver without any valid codeword.

To overcome this, the authors propose a two‑stage decoding pipeline called Complete CA‑SCL (CCA‑SCL). In the first stage the standard soft‑output SCL (SO‑SCL) is run, producing a list of L candidates together with a soft‑output (SO) probability for each candidate. If at least one candidate satisfies the CRC, the candidate with the highest SO is selected, exactly as in conventional CA‑SCL. If the CRC‑passing set is empty, the second stage is invoked: a code‑agnostic outer decoder that can decode the CRC code itself is applied. The outer decoder receives the log‑likelihood ratios (LLRs) corresponding to the CRC‑encoded bits and the CRC parity‑check matrix, and returns a decoded CRC‑codeword together with its own blockwise SO. The outer decoder can be any modern, near‑ML, code‑agnostic algorithm such as GRAND or GCD, preferably in their soft‑output variants (SO‑GRAND, SO‑GCD). By treating the CRC as an error‑correction code rather than a pure detector, the pipeline always outputs a valid codeword, turning CA‑SCL into a complete decoder.

A technical obstacle is the generation of the “outer LLRs” for the CRC bits when a non‑systematic polar code is used. The channel LLRs ℒ_I (associated with the inner polar code bits) must be transformed into LLRs ℒ_O for the CRC bits using a formula derived from the polar transformation matrix. The authors prove two lemmas: (1) the transformed LLRs are generally less reliable than the original channel LLRs, causing performance loss for the outer decoder; (2) the transformed LLRs become statistically correlated, violating the independence assumption of most GRAND/GCD implementations. Consequently, directly applying a code‑agnostic decoder to non‑systematic CA‑Polar yields sub‑optimal gains.

The paper resolves this by employing systematic polar coding. In systematic polar codes the information bits (including the CRC‑encoded bits) appear directly in the transmitted vector, so the channel LLRs for those positions can be used as the outer LLRs without any transformation. This eliminates both the reliability degradation and the correlation issue. Simulations confirm that with systematic CA‑Polar the BLER improvement ranges from 0.2 dB up to 1 dB, whereas the non‑systematic case only gains about 0.2 dB.

Another contribution is the use of soft‑output information to control the undetected error rate (UER). The blockwise SO value S* produced by either SO‑SCL or the outer SO‑GRAND/SO‑GCD can be compared against a threshold 1 − ε. If S* < 1 − ε the decoder declares an erasure, thereby guaranteeing that the probability of an undetected error does not exceed ε. By adjusting ε, the system can trade off BLER against UER in an optimal way, matching the theoretical limits described by Forney’s optimal detection theory.

The overall architecture, illustrated in Figure 1 of the paper, consists of the standard CA‑SCL path (blue lines) and the newly added outer‑decoder and UER‑control paths (dashed lines). The pipeline guarantees a valid codeword for every received block, making it suitable for URLLC where retransmissions are costly. Moreover, the chosen outer decoders (GRAND, GCD) are highly parallelizable and have existing ASIC/FPGA implementations, ensuring that the added complexity can meet the stringent latency requirements of 5G.

Simulation results show:

• For a non‑systematic CA‑Polar code (N = 128, K = 114, M = 90) at Eb/N0 = 6 dB, CCA‑SCL reduces BLER by up to 0.2 dB compared with standard CA‑SCL.
• For the systematic version of the same code, the BLER reduction expands to 0.2–1 dB, confirming the benefit of avoiding LLR conversion.
• UER can be driven down to 10⁻⁶ with negligible BLER impact by setting the SO threshold appropriately.

In conclusion, the authors demonstrate that by augmenting CA‑SCL with a soft‑output, code‑agnostic outer decoder and by leveraging systematic polar coding, the CRC can be repurposed for both error detection and correction. This yields a complete decoder that delivers measurable BLER gains, precise control over undetected errors, and compatibility with existing hardware acceleration techniques—key requirements for next‑generation low‑latency, high‑reliability wireless systems. Future work is suggested on hardware implementation details, dynamic ε adaptation, and extension to fading channels.