Incremental Error Locator Polynomials for Faster Chase Decoding
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Solution Overview
Problem
Chase decoding for algebraic codes is computationally intensive due to its complexity of O(nd) in hardware and software implementations, particularly as code length and minimum Hamming distance increase, necessitating more efficient methods for error correction.
Innovation Solution
Implementing a Chase decoder that generates a new error locator polynomial using interpolation and linear feedback shift register techniques, allowing for parallel processing and reducing complexity to O(n) by avoiding repeated full Berlekamp-Massey processes.
Engineering Contradictions & Design Principles
Engineering Contradiction Analysis
1Reliability
If the straightforward 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 a full Berlekamp-Massey process once to obtain initial error locator polynomials before Chase decoding begins. These pre-computed polynomials serve as starting points for the incremental updates during test error pattern flipping, eliminating the need to repeatedly perform the full Berlekamp-Massey process for each test pattern.
Solution Approach 2:
The patent segments the error locator polynomial generation into two distinct phases: (1) an initial full Berlekamp-Massey process that computes base polynomials, and (2) incremental updates using simple polynomial multiplication and addition operations when flipping test error patterns. This segmentation transforms the computationally intensive repeated full processes into lightweight incremental updates.
2Reliability
If code length and minimum Hamming distance are increased to improve error correction capability, then reliability improves, but decoding time increases significantly
Solution Approach 1:
By pre-computing the error locator polynomials using a full Berlekamp-Massey process before Chase decoding, the patent creates a foundation that enables rapid incremental updates during the actual decoding process. This preliminary computation avoids repeated heavy calculations regardless of code length or minimum Hamming distance.
Solution Approach 2:
The patent implements dynamic polynomial updating where the error locator polynomials are incrementally modified through simple operations (multiplication by monomials and addition) rather than re-computed from scratch. This dynamic approach maintains polynomial accuracy while adapting efficiently to each test error pattern flip.
3Reliability
If hardware or software implementations perform Chase decoding in a straightforward manner, then complete error pattern testing is achieved, but processing speed decreases
Solution Approach 1:
The patent pre-computes error locator polynomials using the Berlekamp-Massey process before initiating Chase decoding. This preliminary computation provides a starting point that enables subsequent rapid incremental updates, significantly accelerating the overall processing speed while maintaining complete error pattern testing.
Solution Approach 2:
The patent segments the decoding process into an initial polynomial computation phase followed by efficient incremental update phases for each test error pattern. This segmentation separates the computationally intensive work from the rapid iterative testing, thereby improving processing speed without sacrificing testing completeness.
Data Source
AI summary
Generating a polynomial is disclosed. A prior error locator polynomial, associated with locating errors in encoded data, is obtained. A new error locator polynomial, associated with a test error pattern, is incrementally generated based at least in part on the prior error locator polynomial.