Systematic TF and PAC Encoding via Triangular Matrix Factorization
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Solution Overview
Problem
There are no methods specifically designed for systematic encoding of Triangular Factorization (TF) codes or Polarization Adjusted Convolutional (PAC) codes, which are newer types of linear block codes, limiting the ability to reduce complexity and improve error correction performance in communication systems.
Innovation Solution
The development of systematic encoding methods for TF codes and PAC codes, utilizing triangular factorization of the generator matrix, which includes invertible upper-triangular and lower-triangular transform matrices, to reduce complexity and enhance error correction capabilities.
Engineering Contradictions & Design Principles
Engineering Contradiction Analysis
1Device complexity
If non-systematic encoding is used for TF codes, then the encoding process is simpler, but the source data does not appear transparently in the encoded form and SER/BER performance deteriorates
Solution Approach 1:
The generator matrix is segmented into triangular factors (L and U matrices) that can be processed independently. The encoding process is divided into two stages: first applying the lower-triangular matrix L to the source data, then applying the upper-triangular matrix U to the intermediate result. This segmentation allows systematic encoding to be achieved without requiring complex full-matrix operations, thus maintaining simplicity while improving SER/BER performance through systematic structure.
Solution Approach 2:
The lower-triangular matrix L is applied to the source data in advance to create an intermediate encoded form that already contains the source data transparently. This preliminary action ensures that the source data bits are preserved in specific positions before the final transformation by matrix U, guaranteeing systematic structure and improved error performance without adding excessive complexity.
2Reliability
If systematic encoding is implemented for TF codes, then the source data appears transparently and SER/BER performance improves, but computational complexity increases
Solution Approach 1:
The complex generator matrix multiplication is segmented into two simpler triangular matrix multiplications. Instead of computing G×s directly with a full generator matrix, the patent computes L×s first (where L is lower-triangular), then U×(L×s) (where U is upper-triangular). This segmentation reduces computational complexity while maintaining systematic encoding properties and improving SER/BER performance.
Solution Approach 2:
The patent inverts the traditional encoding approach by factorizing the generator matrix into triangular components and applying them in reverse order (L first, then U). This inversion allows the source data to appear transparently in the final encoded form while using computationally efficient triangular matrix operations instead of complex full-matrix operations.
3Reliability
If decoding is performed on received code blocks, then error correction is achieved, but energy consumption increases in battery-operated devices
Solution Approach 1:
The systematic encoding structure allows the received code block to self-verify its validity by checking whether the source data portion satisfies the expected systematic structure. If the source data appears transparently and satisfies the systematic form, no decoding operation is needed. This self-service mechanism enables error detection without full decoding, saving energy in battery-operated devices when errors are absent or correctable without decoding.
Data Source
AI summary
A systematic encoder reliably transferring a source data block (SDB) is configured for an outer transform matrix and an inner transform matrix. An inner encoder receives the SDB and generates an output constraint block (OCB) as an SDB image under an inverse of a submatrix of the inner transform matrix. An outer encoder receives a fixed data block (FDB) and the OCB and generates a transform output block (TOB) as a transform input block (TIB) image under the outer transform matrix. The TIB contains the FDB transparently in a sub-block of the TIB, and the TOB contains the OCB transparently in a sub-block of the TOB. The inner encoder receives the TOB and generates a transmitted code block (TCB), transparently containing the SDB in a sub-block therein.