Binary BCH Chase Decoding with Groebner Basis Polynomial Updates
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Solution Overview
Problem
Existing decoding methods for binary BCH codes, such as the Berlekamp-Massey algorithm, are inefficient when handling multiple errors in high-rate codes, requiring significant computational resources and hardware modifications, particularly in NAND flash applications where fast and efficient error correction is crucial.
Innovation Solution
The implementation of a method using an application-specific integrated circuit (ASIC) that performs fast polynomial updates and a modified key equation solution module, leveraging a Groebner basis to reduce computational complexity and unnecessary calculations, thereby improving the efficiency of Chase decoding for binary BCH codes.
Engineering Contradictions & Design Principles
Engineering Contradiction Analysis
1Reliability
If the Berlekamp-Massey algorithm is used for decoding binary BCH codes, then error correction capability is achieved, but computational complexity increases significantly when handling multiple errors in high-rate codes
Solution Approach 1:
The patent segments the decoding process into hard-decision decoding followed by soft-decision refinement only when needed. The Chase decoding algorithm divides errors into correctable errors (handled by HD decoding) and uncorrectable errors (handled by SD refinement), avoiding full SD decoding for all cases and reducing overall computational complexity.
Solution Approach 2:
The patent applies partial soft-decision action by performing syndrome-based SD decoding only for the difference between actual errors and HD correction radius. Instead of always performing full SD decoding, the algorithm selectively applies additional decoding steps only when HD decoding fails to correct all errors, reducing unnecessary computational overhead.
2Productivity
If hard-decision decoding is used for binary BCH codes, then decoding speed is maintained, but decoding fails when errors exceed the correction radius
Solution Approach 1:
The patent introduces soft-decision decoding as an intermediary step between hard-decision decoding and final error correction. When HD decoding identifies uncorrectable errors, the algorithm uses SD decoding with syndrome calculations as a mediator to refine the decision and potentially recover from errors that exceeded the HD correction radius.
Solution Approach 2:
The patent performs preliminary hard-decision decoding first to quickly correct errors within the correction radius before attempting soft-decision refinement. This preliminary action filters out easily correctable cases, allowing the system to maintain high speed for most cases while only applying the more complex SD processing when absolutely necessary.
3Measurement precision
If soft-decision decoding is applied to all received codewords, then decoding accuracy improves, but computational resources and processing time increase
Solution Approach 1:
The patent applies soft-decision decoding partially rather than universally. The algorithm performs SD decoding only for the specific number of errors that exceed the HD correction radius, calculated as (actual errors - HD correction radius). This partial application maintains high accuracy for difficult cases while avoiding the time penalty for cases that HD decoding can handle alone.
Solution Approach 2:
The patent dynamically changes the decoding parameter (decision type) based on the error count. When the number of errors is within the HD correction radius, the system uses hard-decision parameters for fast processing. When errors exceed this radius, the system switches to soft-decision parameters for the necessary additional corrections, optimizing the balance between accuracy and processing time.
4Measurement precision
If polynomial degree grows during Chase decoding, then error locator polynomial accuracy improves, but computational complexity increases due to unnecessary calculations
Solution Approach 1:
The patent extracts and eliminates unnecessary calculation steps from the Chase decoding process. By using the HD decoding results as a starting point and only performing SD syndrome calculations for the difference between actual and correctable errors, the algorithm removes redundant polynomial operations that would otherwise increase degree unnecessarily and consume computational resources.
Solution Approach 2:
The patent performs preliminary hard-decision decoding to establish a baseline error locator polynomial before applying soft-decision refinement. This preliminary action provides an accurate starting point that reduces the degree of polynomial operations needed in the SD phase, avoiding the need to build the complete error locator polynomial from scratch and thereby reducing computational complexity.
Data Source
AI summary
An application specific integrated circuit (ASIC) tangibly encodes a method for fast polynomial updates in fast Chase decoding of binary Bose-Chaudhuri-Hocquenghem (BCH) codes. The method includes the steps of using outputs of a syndrome-based hard-decision (HD) algorithm to find a Groebner basis for a solution module of a modified key equation, upon failure of HD decoding of a BCH codeword received by the ASIC from a communication channel; evaluating polynomials obtained from said Groebner basis at inverses of specified weak-bit locations; and transforming a Groebner basis for a set of flipped weak-bit locations (α1, . . . , αr−1) to a Groebner basis for (α1, . . . , αr), wherein αr is a next weak-bit location, wherein r is a difference between a number of errors and a HD correction radius of the BCH codeword.