Cyclic Code-Based LDPC Decoding with GFT for Soft-Decision RS Codes
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Solution Overview
Problem
Current decoding methods for Reed-Solomon (RS) codes face challenges in efficiently exploiting soft reliability information and achieving low complexity, particularly due to high computational requirements and exponential growth in decoding time with code length, making them impractical for longer codes.
Innovation Solution
A novel coding scheme that maps cyclic codes into quasi-cyclic low-density parity-check (LDPC) codes using Galois Fourier Transform (GFT) and iterative soft decision decoding, allowing for joint encoding and decoding of cyclic codewords to reduce complexity and improve performance.
Engineering Contradictions & Design Principles
Engineering Contradiction Analysis
1Reliability
If traditional algebraic hard-decision decoding methods (BM-HDDA, Euclid's algorithm) are used for Reed-Solomon codes, then decoding complexity is moderate, but error correction performance is insufficient and soft reliability information is not exploited
Solution Approach 1:
The patent introduces an intermediary mapping process that transforms Reed-Solomon code decoding into an equivalent LDPC code decoding problem. By mapping the RS code to an LDPC code through a transformation matrix, the system can utilize iterative belief propagation decoding to achieve better error correction performance while maintaining manageable complexity. The mapping acts as a mediator that bridges the gap between RS code performance requirements and LDPC code decoding capabilities.
Solution Approach 2:
The patent replaces traditional algebraic hard-decision decoding mechanisms with iterative soft-decision belief propagation decoding. Instead of using algebraic algorithms (BM-HDDA, Euclid's algorithm) that cannot exploit soft reliability information, the system substitutes them with message-passing algorithms that process soft information from the channel, thereby improving error correction performance at the cost of increased but manageable computational complexity.
2Reliability
If algebraic soft-decision decoding (KV algorithm) is used to exploit soft reliability information, then error correction performance improves, but decoding complexity increases significantly
Solution Approach 1:
The patent segments the decoding problem by separating the RS code structure from the decoding algorithm. Instead of directly decoding the RS code with complex algebraic soft-decision algorithms, the system transforms the problem into decoding an equivalent LDPC code with a structured parity-check matrix. This segmentation allows the use of simpler iterative belief propagation algorithms while maintaining the ability to exploit soft reliability information, thus reducing decoding complexity compared to direct algebraic soft-decision decoding.
3Reliability
If maximum-likelihood decoding (MLD) based on binary representation is used, then error correction performance reaches optimal level, but running time grows exponentially with code length
Solution Approach 1:
The patent replaces exponential-time maximum-likelihood decoding with polynomial-time iterative belief propagation decoding. By mapping the RS code to an LDPC code and using iterative message-passing algorithms, the system achieves near-MLD performance with manageable computational complexity. The substitution transforms an intractable decoding problem into a practical one that can be solved efficiently even for longer codes.
4Reliability
If turbo decoding or trellis-based decoding is used for RS codes, then soft-decision decoding performance is achieved, but complexity becomes intolerable for practical applications
Solution Approach 1:
The patent introduces an intermediary LDPC code structure that captures the essential properties of RS codes without requiring complex trellis or turbo decoding mechanisms. By mapping the RS code to an LDPC code with a specifically structured parity-check matrix, the system achieves soft-decision decoding performance using simpler iterative algorithms, avoiding the intolerable complexity of trellis-based approaches.
Data Source
AI summary
Techniques are described for joint encoding and decoding of information symbols. In one embodiment, a method for joint encoding includes, in part, obtaining a sequence of information symbols, generating a plurality of cyclic codewords each corresponding to a portion of the sequence of information symbols, jointly encoding the plurality of cyclic codewords to generate at least one combined codeword, and providing the combined codeword to a device. The at least one combined codeword may be generated through Galois Fourier Transform (GFT). In one embodiment, a method for joint decoding includes, in part, obtaining a sequence of encoded symbols, wherein the sequence of encoded symbols is generated through GFT, jointly decoding the sequence of encoded symbols using an iterative soft decision decoding algorithm to generate a decoded sequence, transforming the decoded sequence to generate a plurality of cyclic codewords, and decoding the plurality of cyclic codewords to generate a plurality of decoded information symbols.