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LDPC Encoding and Decoding Techniques


'LDPC' is the abbreviation for 'low density parity check'. LDPC code H matrix contains very few amount of 1's and mostly zeroes.
LDPC codes are error correcting code. Using LDPC codes, channel capacities that are close to the theoretical Shannon limit can be achieved. 
Low density parity check (LDPC) codes are linear error-correcting block code suitable for error correction in a large block sizes transmitted via very noisy channel. Applications requiring highly reliable information transport over bandwidth restrictions in the presence of noise are increasingly using LDPC codes.

1. LDPC Encoding Technique


The proper form of H matrix is derived from the given matrix by doing multiple row operations as shown above.
In the above, H is parity check matrix and G is generator matrix. If you consider matrix H as [-P' | I] then matrix G will be [Ik | P] , where P' is the transpose of P and Ik is the identity matrix of size k by k.

Then the codeword is generated by doing mod 2 multiplication of G times of the message bit string

For example, codeword for bit string '101' is obtained by
( 1 0 1 ) mod 2 multiplication G = ( 1 0 1 0 1 1 )

1.1. Tanner Graph

H * C' = 0
c1Åc2Åc3Åc4 = 0
c3Åc4Åc6 = 0
c1Åc4Åc5 = 0

H (mod 2 multiplication) r = H (mod 2 multiplication) (c + e)
H (mod 2 multiplication) c = 0
H (mod 2 multiplication) e = error

In the above discussion, the H matrix is not  really a low density parity check matrix or sparse matrix.

There are some procedure to form a sparse H matrix.  



2. LDPC Decoding Techniques

Bit Flipping Algorithm for BSC (binary symmetric channel)
For the matrix H above, the parity check set trees can help to correct a bit if it is received in error.

3. Applications of LDPC codes

3GPP 5G-NR
IEEE 802.11n (Wi-Fi)
Wi-max
10 Gbps Ethernet, etc.

4. Advantages of LDPC Codes 

Perform Close to Shannon Limit Capacity
High Throughput

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