Polar Code Construction for Non-Power-of-Two Lengths

The patent embeds polar codes within a larger structured framework by nesting them inside block codes with specific generator matrices. The polar code is constructed as a subset of a block code, where the block code's structure provides additional constraints and properties that enhance performance while maintaining the simplicity of polar code implementation. This nested structure allows the code to benefit from both the capacity-achieving property of polar codes and the performance advantages of structured block codes.

The patent segments the code construction process into distinct stages: first constructing a block code with a specific generator matrix structure, then selectively shortening or puncturing specific positions based on the frozen bit pattern. This segmentation allows for systematic control over code length while maintaining a regular construction procedure. The generator matrix is designed with a specific structure that facilitates easy shortening and puncturing operations without requiring complex re-optimization.

The patent applies local quality by differentiating between different positions in the code based on their error probability characteristics. Frozen bits are placed at positions with higher error probabilities, while information bits are placed at positions with lower error probabilities. When shortening or puncturing is applied, specific positions are selected based on their local characteristics rather than uniformly affecting all positions. This localized approach ensures that the most critical parts of the code maintain their error correction capabilities even when the code length is reduced.