Generator Matrix Triangular Factorization for Low-Complexity TF Codes
Find Innovative SolutionsGenerate Solutions
Solution Overview
Problem
Polar codes face inferior performance at practical block-lengths due to limitations in error correction capabilities and high complexity in existing decoding methods, failing to compete with prior art in error correction coding.
Innovation Solution
The introduction of triangular factorization (TF) codes, which utilize a non-trivial triangular factorization of the code generator matrix to create channel polarization and facilitate efficient decoding through tree-search algorithms, avoiding the complexity of traditional concatenated coding schemes.
Engineering Contradictions & Design Principles
Engineering Contradiction Analysis
1Reliability
If traditional concatenated coding schemes are used to improve error correction performance, then frame error rate performance improves, but device complexity increases significantly
Solution Approach 1:
The generator matrix G is segmented into two triangular factors G = G1 * G2, where G1 is upper triangular and G2 is lower triangular. This segmentation allows the encoding process to be divided into two simpler stages, each with reduced computational complexity compared to the original concatenated coding scheme, while maintaining the error correction performance benefits.
Solution Approach 2:
The patent introduces an intermediate transformed block u that serves as a mediator between the input data block and the final encoded output. This intermediate block allows the encoding process to proceed through two triangular transformations rather than requiring complex concatenated coding operations, thereby reducing overall device complexity while preserving reliability.
2Device complexity
If polar codes are used to achieve channel capacity with low complexity encoding, then encoding complexity is reduced, but error correction capability becomes insufficient at practical block-lengths
Solution Approach 1:
The patent creates a composite coding structure by combining two different triangular transforms (G1 and G2) into a unified generator matrix. This composite approach leverages the advantages of both transforms to achieve superior error correction capability at practical block-lengths while maintaining low encoding complexity, effectively creating a hybrid code that outperforms standard polar codes.
3Reliability
If non-trivial triangular factorization is applied to the generator matrix to create channel polarization, then error correction capability improves, but computational complexity increases
Solution Approach 1:
The patent changes the parameters of the transformation matrices by specifically designing G1 and G2 with particular structural properties (upper and lower triangular forms with specific non-zero patterns). These parameter changes enable the factorization to create effective channel polarization for improved error correction while keeping the computational operations simple and efficient through the triangular structure.
Data Source
AI summary
An encoder apparatus for reliable transfer of a source data block d in a communication system includes an outer transform configured to receive a data container block v and compute an outer transform block u, whereby u=vGout for an outer transform matrix Gout. The encoder apparatus also includes an inner transform configured to receive the outer transform block u and compute a transmitted code block x, whereby x=uGin for an inner transform matrix Gin. The data container block v is obtained from the source data block d and a frozen data block a. The frozen data block a is a predetermined block of symbols. The outer transform matrix Gout and the inner transform matrix form a triangular factorization of a transform matrix G, which optionally is a non-triangular matrix, while the outer transform matrix Gout and the inner transform matrix Gin are strictly upper- and lower-triangular matrices, respectively.