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Low-density parity-check (LDPC) codes are a class of highly efficient linear block codes made from many single parity check (SPC) codes. They can provide performance very close to the channel capacity (the theoretical maximum) using an iterated soft-decision decoding approach, at linear time complexity in terms of their block length.
In this example, the first bit cannot yet be recovered, because all of the constraints connected to it have more than one unknown bit. In order to proceed with decoding the message, constraints connecting to only one of the erased bits must be identified. In this example, only the second constraint suffices.
For example, p 2 provides an even parity for bits 2, 3, 6, and 7. It also details which transmitted bit is covered by which parity bit by reading the column. For example, d 1 is covered by p 1 and p 2 but not p 3 This table will have a striking resemblance to the parity-check matrix (H) in the next section.
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All error-detection and correction schemes add some redundancy (i.e., some extra data) to a message, which receivers can use to check consistency of the delivered message and to recover data that has been determined to be corrupted.
• Length. The length is the number of evaluation points. Because the sets are disjoint for {, …,}, the length of the code is | | = (+). • Dimension. The dimension of the code is (+), for ≤ , as each has degree at most (()), covering a vector space of dimension (()) =, and by the construction of , there are + distinct .
Some important terminologies & constraints :- Implicit Numbering : It simply means that nodes are numbered in increasing order by level and from left to right. i.e. nodes at bottom level will have low implicit number as compared to upper level nodes and nodes on same level are numbered in increasing order from left to right.
Proof. We need to prove that if you add a burst of length to a codeword (i.e. to a polynomial that is divisible by ()), then the result is not going to be a codeword (i.e. the corresponding polynomial is not divisible by ()).