Protograph LDPC Encoding for Low-Rate Error Correction
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Solution Overview
Problem
Existing low-density parity-check (LDPC) codes face challenges in achieving high girth and efficient error correction for low-rate codes, particularly in terms of complexity and performance, especially for moderate codeword lengths and non-binary symbol dimensions.
Innovation Solution
A protograph-based encoding scheme is introduced, utilizing non-binary symbols from a Galois Field of prime or power of prime dimension, with v-nodes grouped into three parallel paths and c-nodes of specific degrees, optimized for maximum girth and Hamming distance, and employing interleaving and non-zero value assignment for improved error correction.
Engineering Contradictions & Design Principles
Engineering Contradiction Analysis
1Reliability
If conventional LDPC codes are used for low-rate coding, then error correction capability is improved, but implementation complexity increases
Solution Approach 1:
The code is segmented into a mother code with rate 1/3 and punctured versions at lower rates. The encoder structure is divided into systematic and parity parts, with the parity encoder further divided into sub-units that can be selectively activated. This segmentation allows the system to achieve low-rate coding through simple puncturing rather than designing entirely new complex codes for each rate.
Solution Approach 2:
A single mother code at rate 1/3 serves multiple functions: it can be used directly at rate 1/3 or punctured to generate codes at lower rates (1/6, 1/9, etc.). The same encoder hardware can operate at different rates by simply disabling certain parity sub-units, eliminating the need for multiple specialized encoders for different coding rates.
2Reliability
If code rate is reduced for better error correction, then reliability is improved, but coding overhead increases
Solution Approach 1:
The mother code is designed in advance with a specific structure where parity bits are organized in sub-units that can be systematically punctured. This preliminary structuring allows for efficient rate adaptation without requiring additional complex processing or overhead when transitioning to lower rates.
3Reliability
If protograph structure is optimized for maximum girth, then error floor is eliminated, but design complexity increases
Solution Approach 1:
The protograph parameters are specifically chosen: variable nodes have degree 2, check nodes have degrees 2 or 3, and the graph structure is arranged to maximize girth. These parameter choices systematically eliminate short cycles that cause error floors while maintaining a relatively simple regular structure that is easier to design than completely irregular high-girth graphs.
4Reliability
If non-binary symbols are used for moderate codeword lengths, then decoding performance is improved, but computational complexity increases
Solution Approach 1:
The binary LDPC code structure is copied and extended to non-binary symbols from Galois fields GF(q). The same protograph structure and decoding algorithms are applied, but operations are performed in the extended non-binary field. This allows leveraging the well-understood binary code design methodology while gaining the performance benefits of non-binary symbols for moderate-length codes.
Data Source
AI summary
An encoder and a decoder employ an encoding scheme corresponding to a parity check matrix which is derivable from a bipartite protograph formed of variable nodes and check nodes, with each variable node corresponding to a codeword symbol position. The protograph has a plurality of groups of nodes, each group of nodes comprising both variable nodes and check nodes. Each of the check nodes in a group is of degree 2 and has connections to two variable nodes in the same group. The protograph also has a plurality of check nodes of degree n, where n is the number of said plurality of groups, wherein each of the plurality of check nodes has a connection to a variable node in each group such that the symbol positions in a codeword are interleaved between the groups of nodes.