BCH Code Decoding via Binary Syndrome Matrix Computation
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Solution Overview
Problem
Existing methods for decoding block codes, such as BCH codes, are inefficient in rapidly processing communication signals and correcting errors, particularly in scenarios where error-free blocks are common, as they require extensive computations and large tables.
Innovation Solution
A method that rapidly computes syndrome sequences for binary BCH codes by using a binary-matrix multiplication approach, converting the syndrome into a syndrome sequence, which allows efficient error calculation without large tables, and involves establishing a generator polynomial, check polynomial, and binary syndrome calculation to correct errors in communication signals.
Engineering Contradictions & Design Principles
Engineering Contradiction Analysis
1Productivity
If traditional syndrome-based decoding methods are used for BCH codes, then error correction capability is maintained, but decoding speed and computational efficiency deteriorate due to extensive computations and large tables required
Solution Approach 1:
The patent segments the syndrome computation process into distinct binary matrix multiplication operations. By dividing the complex syndrome calculation into smaller matrix operations (computing S = H·r^T where H is the parity-check matrix), the method enables more efficient parallel processing and reduces the computational burden compared to traditional sequential syndrome-based decoding methods
Solution Approach 2:
The patent replaces traditional mechanical computation methods (extensive sequential computations and large lookup tables) with a matrix-based algebraic approach. By using binary matrix multiplication and polynomial representations, the method substitutes complex iterative calculations with more efficient linear algebra operations that can be executed faster and with less computational resources
2Reliability
If traditional syndrome-based decoding methods are used for BCH codes, then error correction capability is maintained, but processing time increases due to extensive computations
Solution Approach 1:
The patent performs preliminary setup of the binary parity-check matrix H and generator polynomial g(x) before actual decoding operations. By pre-computing and storing the matrix structure and polynomial coefficients, the method eliminates the need for repeated complex calculations during actual decoding, significantly reducing processing time while maintaining error correction capability
Solution Approach 2:
The patent uses polynomial representations and binary matrix copies to represent the code structure. By working with copied polynomial forms and matrix representations rather than directly manipulating the original complex syndrome data, the method enables faster computation and reduces decoding time while preserving the mathematical properties needed for error correction
Data Source
AI summary
A method of correcting a communication signal with BCH product codes is disclosed. The method comprising the steps of receiving a codeword vector, establishing a generator polynomial, establishing a check polynomial, calculating a binary-matrix, and calculating the binary syndrome S=Hrt. If an all zero vector results, no errors exist in the received vector. If errors exist, a tm x (N−k) binary matrix is created, the power basis expansion of (r(α1), r(α3), r(α5), . . . r(α2t−1)) is calculated, and the syndrome is solved. The speed of the method comes from using word-level XOR's to apply the check polynomial to the stream at all offsets. The utility of this approach is due in part to the fact that the conversion matrix involved can be created quickly using only items needed elsewhere in BCH decoders: a field table and the generator polynomial coefficients.