GRS Chase Decoding with Groebner Bases for Faster Error Correction
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Solution Overview
Problem
Existing decoding algorithms for generalized Reed-Solomon codes, such as Wu's fast Chase decoding algorithm, are complex and not capable of handling a total number of errors larger than d−1, requiring exhaustive root searches and being difficult to implement in frequency domains.
Innovation Solution
A new fast Chase algorithm for generalized Reed-Solomon codes that uses a syndrome-based hard-decision method, employing Koetter's algorithm to find a Groebner basis for error locations and incorporating a stopping criterion to avoid unnecessary Chien searches, allowing for operation with more than d−1 errors and simplifying the decoding process.
Engineering Contradictions & Design Principles
Engineering Contradiction Analysis
1Speed
If Wu's fast Chase decoding algorithm is used, then decoding speed is improved, but the algorithm complexity increases and it cannot handle more than d-1 errors
Solution Approach 1:
The patent segments the error handling process by separating the identification of error locations (using modified Chien search) from error value calculation (using Forney's formula). It also segments the syndrome computation into distinct phases and uses intermediate results to avoid redundant calculations, thereby reducing overall algorithm complexity while maintaining speed.
Solution Approach 2:
The patent performs preliminary syndrome computation and error location identification before final error value calculation. By pre-computing syndromes and identifying error locations upfront, the algorithm avoids redundant operations during the main decoding process, reducing complexity while preserving decoding speed.
2Reliability
If exhaustive root searches are performed, then all error locations are found, but the decoding complexity increases significantly
Solution Approach 1:
The patent applies a modified Chien search that performs partial root searching rather than exhaustive searches. By using the syndrome information to guide the search and stopping criteria to terminate early when sufficient error locations are found, it achieves reliable error detection without the full complexity of exhaustive searches.
3Ease of manufacture
If the algorithm is designed for syndrome-based hard-decision decoding, then implementation is simplified, but it cannot operate effectively in frequency domains
Solution Approach 1:
The patent designs the algorithm with universal applicability by formulating the syndrome computation and error correction steps in a way that can operate in both time domain (syndrome-based hard-decision) and frequency domain. The use of polynomial representations and modular arithmetic allows the same algorithmic structure to function in different domains, enhancing versatility without sacrificing implementation simplicity.
4Reliability
If the error correction capability is increased beyond d-1, then more errors can be handled, but the existing algorithms become inapplicable
Solution Approach 1:
The patent changes key parameters of the decoding algorithm, particularly the syndrome computation formulas and error location identification methods, to accommodate error counts beyond d-1. By modifying the mathematical relationships and using generalized forms of the key equations, the algorithm maintains validity and correctness for higher error correction capabilities.
Data Source
AI summary
An application specific integrated circuit (ASIC) tangibly encodes a program of instructions executable by the integrated circuit to perform a method for fast Chase decoding of generalized Reed-Solomon (GRS) codes. The method includes using outputs of a syndrome-based hard-decision (HD) algorithm to find an initial Groebner basis G for a solution module of a key equation, upon failure of HD decoding of a GRS codeword received by the ASIC from a communication channel; traversing a tree of error patterns on a plurality of unreliable coordinates to adjoin a next weak coordinate, where vertices of the tree of error patterns correspond to error patterns, and edges connect a parent error pattern to a child error pattern having exactly one additional non-zero value, to find a Groebner basis for each adjoining error location; and outputting an estimated transmitted codeword when a correct error vector has been found.