[Paper Review] Recursive Descriptions of Decoding Algorithms and Hardware Architectures for Polar Codes

Noam Presman, Simon Litsyn|arXiv (Cornell University)|Sep 21, 2012

Coding theory and cryptography3 citations

TL;DR

This paper presents a recursive formalization of successive cancellation (SC), list successive cancellation (SCL), and belief propagation (BP) decoding algorithms for polar codes, enabling systematic design for arbitrary polarizing kernels. It introduces recursive hardware architectures that simplify implementation and extend to both standard and custom kernels, enhancing flexibility and scalability in practical deployments.

ABSTRACT

Polar codes are recursive general concatenated codes. This property motivates a recursive formalization of the known decoding algorithms: Successive Cancellation, Successive Cancellation with Lists and Belief Propagation. This description allows an easy development of the first two algorithms for arbitrary polarizing kernels. Hardware architectures for these decoding algorithms are also described in a recursive way, both for Arikan's standard polar codes and for arbitrary polarizing kernels.

Motivation & Objective

• To formalize decoding algorithms for polar codes using a recursive framework to improve algorithmic clarity and extensibility.
• To enable the design of hardware architectures for decoding algorithms that are scalable and reusable across different polarizing kernels.
• To extend existing decoding methods—SC, SCL, and BP—to arbitrary polarizing kernels through a unified recursive description.
• To provide a systematic methodology for hardware implementation that reduces design complexity and supports future code optimizations.

Proposed method

• The paper introduces a recursive description of the successive cancellation (SC) decoding algorithm using a tree-like structure based on the polarizing kernel's recursive decomposition.
• It extends the recursive formalization to list successive cancellation (SCL) decoding by incorporating multiple paths in the recursion, maintaining the same structural framework.
• Belief propagation (BP) decoding is also recast in a recursive form, aligning it with the same recursive kernel structure for consistency and modularity.
• Hardware architectures are derived recursively from the algorithmic descriptions, enabling modular and scalable designs for both Arikan's standard kernel and arbitrary kernels.
• The recursive hardware design supports parameterized implementation, allowing reuse across different code lengths and kernel types.
• The approach enables automatic generation of hardware schematics from the recursive algorithmic description, reducing manual design effort.

Experimental results

Research questions

• RQ1 How can decoding algorithms for polar codes be formally described in a recursive manner to support arbitrary polarizing kernels?
• RQ2 What is the impact of recursive algorithmic formalization on the design and modularity of hardware decoders?
• RQ3 Can recursive hardware architectures be derived directly from recursive algorithm descriptions to ensure consistency and scalability?
• RQ4 How does the recursive framework enable extension to non-standard polarizing kernels beyond Arikan's original construction?
• RQ5 What are the benefits of this recursive approach in terms of hardware resource reuse and design maintainability?

Key findings

• The recursive formalization enables a unified and extensible description of SC, SCL, and BP decoding algorithms for any polarizing kernel.
• The recursive hardware architecture design reduces implementation complexity and supports automatic generation of circuit diagrams from algorithmic descriptions.
• The approach allows seamless extension from standard Arikan kernels to arbitrary polarizing kernels without redesigning the core decoding logic.
• The recursive framework enhances modularity, enabling reuse of design components across different code lengths and kernel types.
• The method provides a scalable path for hardware deployment of advanced polar code decoding beyond conventional implementations.