Incremental Error Locator Polynomial Updates for Chase Decoding
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Solution Overview
Problem
Chase decoding for Reed-Solomon codes is computationally intensive due to its complexity of O(nd) with respect to code length (n) and minimum Hamming distance (d), making it inefficient in hardware and software implementations as code length and Hamming distance increase.
Innovation Solution
The implementation of an incremental generation method for error locator polynomials using interpolation and linear feedback shift registers, which reduces the complexity to O(n) by generating new polynomials without repeatedly performing the full Berlekamp-Massey process, and allows for parallel processing of multiple interpolation-based linear feedback shift registers.
Engineering Contradictions & Design Principles
Engineering Contradiction Analysis
1Reliability
If the full Berlekamp-Massey process is repeatedly performed for each test error pattern in Chase decoding, then accurate error correction is achieved, but the computational complexity increases to O(nd)
Solution Approach 1:
The patent performs the Berlekamp-Massey process once to obtain an initial error locator polynomial before Chase decoding begins. This preliminary computation provides a starting point that eliminates the need to repeatedly execute the full Berlekamp-Massey algorithm for each test error pattern, thereby reducing computational complexity while maintaining decoding accuracy
Solution Approach 2:
The patent segments the error locator polynomial into components that can be efficiently updated using linear feedback shift registers during Chase decoding. By dividing the polynomial processing into manageable segments that can be incrementally modified, the system avoids the O(nd) complexity of full repeated Berlekamp-Massey executions
2Reliability
If code length and minimum Hamming distance are increased to improve error correction capability, then reliability improves, but decoding time increases
Solution Approach 1:
By pre-computing the error locator polynomial using the Berlekamp-Massey process before Chase decoding, the patent establishes a foundation that allows subsequent decoding operations to proceed more quickly. This preliminary action is particularly beneficial for codes with larger minimum Hamming distances, where the reduction in repeated computations translates to significant decoding time savings
Solution Approach 2:
The patent replaces the mechanical repetition of the full Berlekamp-Massey algorithm with a more efficient computational approach using linear feedback shift registers and polynomial updates. This substitution maintains the ability to handle codes with increased length and Hamming distance while avoiding the quadratic time complexity that would otherwise result
3Productivity
If hardware implementations are used to improve processing speed, then productivity increases, but device complexity and cost increase
Solution Approach 1:
The patent segments the polynomial processing into modular components that can be efficiently implemented using linear feedback shift registers in hardware. This segmentation allows for standardized, reusable hardware modules that reduce overall system complexity while maintaining high processing speeds
Solution Approach 2:
The patent designs the hardware architecture to perform multiple functions using the same core components. The linear feedback shift registers and polynomial update mechanisms serve both the initial Berlekamp-Massey process and the subsequent Chase decoding operations, reducing the need for separate dedicated hardware for each function and thereby lowering overall hardware complexity
Data Source
AI summary
An error locator polynomial is incrementally generated by flipping a bit pattern Yi at a symbol Xi an initial dataword to obtain a first test error pattern. A bit pattern Yj at a symbol Xj within the first test error pattern is flipped to obtain a second test error pattern, wherein i≠j.