List Decoding of Polar Codes (1206.0050v1)

Abstract: We describe a successive-cancellation \emph{list} decoder for polar codes, which is a generalization of the classic successive-cancellation decoder of Ar{\i}kan. In the proposed list decoder, up to $L$ decoding paths are considered concurrently at each decoding stage. Then, a single codeword is selected from the list as output. If the most likely codeword is selected, simulation results show that the resulting performance is very close to that of a maximum-likelihood decoder, even for moderate values of $L$. Alternatively, if a "genie" is allowed to pick the codeword from the list, the results are comparable to the current state of the art LDPC codes. Luckily, implementing such a helpful genie is easy. Our list decoder doubles the number of decoding paths at each decoding step, and then uses a pruning procedure to discard all but the $L$ "best" paths. %In order to implement this algorithm, we introduce a natural pruning criterion that can be easily evaluated. Nevertheless, a straightforward implementation still requires $\Omega(L \cdot n^2)$ time, which is in stark contrast with the $O(n \log n)$ complexity of the original successive-cancellation decoder. We utilize the structure of polar codes to overcome this problem. Specifically, we devise an efficient, numerically stable, implementation taking only $O(L \cdot n \log n)$ time and $O(L \cdot n)$ space.

• The paper introduces a successive-cancellation list decoder that evaluates multiple paths concurrently to achieve near-ML performance with moderate list sizes.
• The paper presents an efficient implementation that reduces decoding complexity to O(L·n log n) by leveraging polar codes' structure and lazy-copy techniques.
• The paper demonstrates that augmenting polar codes with a CRC concatenation scheme significantly improves error correction in practical communication systems.

List Decoding of Polar Codes

The paper "List Decoding of Polar Codes" by Ido Tal and Alexander Vardy presents a significant extension to the decoding methodology for polar codes, originally developed by Arikan. The central contribution is the development of a successive-cancellation list (SCL) decoder, which performs concurrent decoding along multiple paths (up to a maximum of $L$ paths), as opposed to the single-path approach of the classic successive-cancellation (SC) decoder.

Overview and Key Contributions

1. Generalization of SC Decoder:
   • The paper introduces an SCL decoder where $L$ decoding paths are concurrently evaluated at each stage.
   • A single codeword is selected from these paths, aiming to choose the most probable one.
   • Simulation results indicate that this method approximates maximum-likelihood (ML) decoding, even with moderate values of $L$.
2. Implementation Constraints:
   • Although the naive implementation has a time complexity of $\Omega(L \cdot n^2)$, this complexity is greatly reduced using the structure of polar codes.
   • The authors propose an efficient implementation achieving a complexity of $O(L \cdot n \log n)$ in time and $O(L \cdot n)$ in space.

Main Algorithmic Details

The SCL decoding functions essentially by facilitating path proliferation and pruning:

• Path Duplication and Pruning:
  • At each decoding step, the number of paths is doubled.
  • Only the $L$ most promising paths are retained, based on a natural pruning criterion that is computationally manageable.
• Efficient Implementation:
  • Through careful management of data structures, the authors optimize space and time complexity. Specifically, the lazy-copy methodology ensures that the copying of large data structures is minimized.

Numerical Results

Simulation Results:

• The paper provides simulation data for a block length of $n=2048$ and rate $1/2$. For various list sizes $L$, the results are plotted in terms of the word error rate (WER).
• The simulations show that with a carefully chosen path (most probable path), the decoder performance closely matches that of the ML decoder.
• Performance with Larger $L$:
  • An increase in $L$ enhances performance, but with diminishing returns beyond a certain point. For example, moving from $L = 16$ to $L = 32$ yields a smaller gain compared to moving from $L = 1$ to $L = 16$.

Modified Polar Codes

The authors introduce an enhancement through the use of a CRC concatenation scheme:

• Concatenation Scheme:
  • Add a small CRC to the information bits and use the SCL decoder to verify the integrity of the path picked.
  • This modification, though slightly altering the polar code, significantly improves the decoder's performance by correctly leveraging the linear property of CRC.
• Simulation with CRC:
  • The results demonstrate remarkable improvements in error rates by using CRC, contributing to nearly optimal performance even in practical scenarios.

Implications and Future Directions

Practical Implications:

• The proposed decoding methodology provides practical performance close to that of ML decoders while being computationally feasible.
• It represents a step forward for applications requiring efficient and robust error correction mechanisms, such as 5G communication systems.

Theoretical Contributions:

• This work bridges the gap between theoretical capacity-achieving codes and practical implementations.
• It also opens avenues for further research into improved decoding algorithms that balance complexity and performance.

Future Work:

• The prospects of extending this approach to non-binary and non-symmetric channels.
• Implementing adaptive list sizes to dynamically optimize performance based on channel conditions.

In conclusion, Tal and Vardy's "List Decoding of Polar Codes" advances the state of polar codes by introducing an SCL decoder that closely approximates ML performance while being computationally efficient. The enhancement using CRC demonstrates practical viability, making this a valuable contribution to both coding theory and its applications.