'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
c1ร
c2ร
c3ร
c4 = 0
c3ร
c4ร
c6 = 0
c1ร
c4ร
c5 = 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.
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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