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Formally, a parity check matrix H of a linear code C is a generator matrix of the dual code, C ⊥. This means that a codeword c is in C if and only if the matrix-vector product Hc ⊤ = 0 (some authors [1] would write this in an equivalent form, cH ⊤ = 0.) The rows of a parity check matrix are the coefficients of the parity check equations. [2]
A self-dual code is one which is its own dual. This implies that n is even and dim C = n/2.If a self-dual code is such that each codeword's weight is a multiple of some constant >, then it is of one of the following four types: [1]
In coding theory, an expander code is a [,] linear block code whose parity check matrix is the adjacency matrix of a bipartite expander graph.These codes have good relative distance (), where and are properties of the expander graph as defined later, rate (), and decodability (algorithms of running time () exist).
For practical purposes, parity-check matrix of a binary Goppa code is usually converted to a more computer-friendly binary form by a trace construction, that converts the -by-matrix over () to a -by-binary matrix by writing polynomial coefficients of () elements on successive rows.
where H is the parity-check matrix of the Hamming code and is given by = []. The [[,,]] Steane code is the first in the family of ...
Indirect parity measurements coincide with the typical way we think of parity measurement as described above, by measuring an ancilla qubit to determine the parity of the input bits. Direct parity measurements differ from the previous type in that a common mode with the parities coupled to the qubits is measured, without the need for an ancilla ...
In the linear code case a different proof of the Singleton bound can be obtained by observing that rank of the parity check matrix is . [4] Another simple proof follows from observing that the rows of any generator matrix in standard form have weight at most n − k + 1 {\displaystyle n-k+1} .
There are two separate issues here: the origin of good parity-check matrices, and the current "best" performance. I agree with Nahaj that there is more than one way of creating a good parity-check matrix; some documentation of good methods needs to be documented. The LDPC world splits slightly into two camps, psuedo-random generation and non ...