Cyclic Code Joint Decoding via LDPC Mapping and GFT
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Solution Overview
Problem
Existing decoding methods for Reed-Solomon (RS) codes face high computational complexity and inefficiency due to the exponential growth of decoding time with code length, making them impractical for lengths beyond 15, and are not suitable for RS codes due to high-density parity-check matrices and short cycles in their Tanner graphs.
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 Hadamard permutations, allowing for iterative soft-decision decoding with reduced complexity and improved performance by transforming the received sequence into a binary LDPC code with a circulant permutation matrix structure.
Engineering Contradictions & Design Principles
Engineering Contradiction Analysis
1Reliability
If traditional algebraic decoding methods (BM-HDDA, Euclid's algorithm) are used for Reed-Solomon codes, then decoding can be performed with moderate complexity, but decoding performance shows little improvement for practical signal-to-noise ratio values and soft reliability information is not exploited
Solution Approach 1:
The patent introduces an intermediary transformation process that converts Reed-Solomon code decoding into an equivalent LDPC code decoding problem. By mapping the RS code parity-check equations into an LDPC framework through coordinate transformations and permutation operations, the solution enables the use of soft-decision belief propagation algorithms while maintaining computational tractability. This intermediary approach allows exploitation of soft reliability information without directly implementing complex algebraic soft-decision algorithms.
Solution Approach 2:
The patent transforms the decoding problem by changing the parameter representation of the code structure. Specifically, it converts the high-density parity-check matrix of RS codes into a sparse LDPC parity-check matrix through parameter transformations including coordinate permutations and field element mappings. This parameter change enables the application of iterative belief propagation algorithms that can effectively utilize soft reliability information from the channel.
2Reliability
If maximum-likelihood decoding is applied to Reed-Solomon codes based on binary representation as union of cosets, then MLD performance is achieved, but running time grows exponentially with code length
Solution Approach 1:
The patent segments the Reed-Solomon code into an equivalent LDPC code structure with sparse parity-check matrices. By representing the RS code as an LDPC code with a specific Tanner graph structure, the decoding problem is divided into local computations at check nodes and variable nodes, avoiding the exponential complexity of global maximum-likelihood decoding. This segmentation enables polynomial-time iterative decoding while approaching MLD performance.
Solution Approach 2:
The patent substitutes the mechanical exhaustive search required for maximum-likelihood decoding with an iterative belief propagation mechanism. Instead of evaluating all possible codewords (mechanical enumeration), the system uses message passing algorithms that iteratively refine probability estimates, replacing the computationally intensive mechanical system with an efficient probabilistic inference system.
3Reliability
If iterative belief propagation decoding is directly applied to Reed-Solomon codes, then soft reliability information can be exploited, but high-density parity-check matrices and short cycles in Tanner graphs make direct application unsuitable
Solution Approach 1:
The patent introduces asymmetry in the representation of Reed-Solomon codes by mapping them to LDPC codes with asymmetric parity-check structures. The transformation creates a Tanner graph with specific asymmetric properties that reduce short cycles while maintaining error-correcting capability. This asymmetric representation enables belief propagation to converge effectively by breaking the symmetry that causes problematic short cycles in direct RS code representations.
Solution Approach 2:
The patent transitions from the traditional algebraic representation of Reed-Solomon codes to a graphical model representation in another dimension (the Tanner graph of an equivalent LDPC code). This dimensional change from algebraic equations to graphical message-passing structures enables the application of belief propagation algorithms, adding a new dimension to the decoding approach that facilitates soft-decision processing while avoiding the short cycle problems of direct representations.
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.