Polar Code Partial Repetition for Low-Weight Codeword Reduction
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Solution Overview
Problem
Polar codes exhibit a 1 decibel gap in performance for finite block lengths due to poor weight distribution, particularly with minimum weight rows, which affects their error-correcting capabilities in wireless communication.
Innovation Solution
The introduction of partial repetition in the polar encoding process, where specific bits are repeated from the repeated set to the repeating set based on weight thresholds, reduces the number of minimum weight codewords without increasing decoding complexity, enhancing error-correcting performance.
Engineering Contradictions & Design Principles
Engineering Contradiction Analysis
1Reliability
If polar codes are used for wireless communication, then encoding and decoding can be performed efficiently, but error-correcting performance is degraded due to poor weight distribution with minimum weight rows
Solution Approach 1:
The information set is segmented into multiple subsets based on weight thresholds. Indices are divided into different groups (first set, second set, third set) according to their row weights in the generator matrix. This segmentation allows selective repetition of bits from specific weight categories to improve error correction without processing all bits uniformly.
Solution Approach 2:
The patent changes the parameter of bit repetition by introducing a weight threshold parameter. Bits are repeated based on their associated row weights in the generator matrix, with different repetition patterns applied to bits from different weight categories. This parameter-based approach transforms the uniform coding structure into a weight-adaptive structure that improves minimum distance properties.
2Reliability
If bits are repeated to reduce minimum weight codewords, then error-correcting ability improves, but coding complexity increases
Solution Approach 1:
Different repetition operations are applied locally to different bit positions based on their weight characteristics. Instead of uniformly repeating all bits, the patent applies repetition selectively to bits from specific weight categories (e.g., repeating bits from the first set differently from bits in the second set). This local differentiation improves error correction where needed while minimizing unnecessary operations elsewhere.
Solution Approach 2:
The patent performs partial repetition rather than complete repetition of all information bits. Only specific subsets of bits (those falling into certain weight categories) are repeated, and the repetition is performed partially (not all bits from each subset necessarily repeat). This partial action achieves the necessary error correction improvement without the full complexity cost of repeating every bit.
3Reliability
If partial repetition is applied to reduce low-weight codewords, then performance gap closes by 0.1 decibels, but the encoding process becomes more complex
Solution Approach 1:
The patent performs preliminary classification of bit indices into weight-based subsets before the actual encoding process. By pre-organizing indices into sets based on their row weights in the generator matrix, the system prepares the structure needed for selective repetition in advance. This preliminary action simplifies the subsequent encoding process by having the categorization already done, rather than making weight calculations during encoding.
Data Source
AI summary
Error-correcting performance of polar codes is improved by reducing low-weight codewords through partial repetition. A threshold for weights of rows with indices belonging to an information set is obtained in a polar coding generator matrix, and a smallest index of the information set with a weight less than or equal to the threshold is obtained. A first candidate index is selected based on a frozen index that is larger than the smallest index, and a second candidate index is selected based on information indices that are smaller than a largest candidate frozen index and that corresponds to a weight less than or equal to the threshold. Support of the first candidate index is determined to be distinct from support of the second candidate index in at least two elements, based on which at least one bit from the second candidate index is repeated to the first candidate index.