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Low-Density Parity Check (LDPC)
Low-Density Parity Check (LDPC) codes are linear error-correcting block codes designed for correcting errors in large block sizes transmitted through very noisy channels. These codes provide excellent error correction performance while maintaining relatively low computational complexity.
LDPC codes were developed by Robert G. Gallager in his doctoral dissertation at the Massachusetts Institute of Technology in 1960. Due to their inventor, these codes are also known as Gallager codes.
Structure of LDPC Codes
An LDPC code is specified by a parity-check matrix that contains mostly 0s and a low density of 1s. The rows of this matrix represent the parity equations, while the columns represent the bits in the codeword (code symbols).
An LDPC code is represented as (n, j, k), where:
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n is the block length (total number of bits in the codeword)
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j is the number of 1s in each column, where j ? 3
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k is the number of 1s in each row, where k > j
LDPC vs Hamming Code Comparison
| Feature | Hamming Code | LDPC Code |
|---|---|---|
| Matrix Structure | Diagonal 1s in parity bits | No diagonal structure |
| Density of 1s | Higher density | Low density (sparse) |
| Block Size | Small blocks | Large blocks |
| Performance | Good for small errors | Excellent for noisy channels |
Decoding Techniques
LDPC codes can be decoded using two main approaches:
Hard Decision Decoding
The decoder performs all parity checks according to the parity equations. If any bit appears in more than a fixed number of unsatisfied parity equations, its value is flipped. The process repeats until all parity equations are satisfied. This method is simple but works best with small parity-check sets.
Soft Decision Decoding
This method uses probabilistic algorithms on LDPC graphs ? sparse bipartite graphs with two node sets: one representing parity equations and another representing code symbols. Lines connect nodes when a code symbol appears in an equation.
Decoding involves passing messages between nodes. The two main subclasses are belief propagation and maximum likelihood decoding. Though computationally complex, these algorithms provide superior error correction performance.
Conclusion
LDPC codes offer exceptional error correction capabilities for large block transmissions over noisy channels. Their sparse parity-check matrix structure enables efficient decoding algorithms, making them ideal for modern communication systems requiring high reliability and throughput.
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