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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]
As a result, once reduction is complete the parity errors sometimes seen on the 4×4×4 cannot occur on the 5×5×5, or any cube with an odd number of layers. [9] The Yau5 method is named after its proposer, Robert Yau. The method starts by solving the opposite centers (preferably white and yellow), then solving three cross edges (preferably ...
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.
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 other words, the matrix of the combined transformation A followed by B is simply the product of the individual matrices. When A is an invertible matrix there is a matrix A −1 that represents a transformation that "undoes" A since its composition with A is the identity matrix. In some practical applications, inversion can be computed using ...
A wide range of datasets are naturally organized in matrix form. One example is the movie-ratings matrix, as appears in the Netflix problem: Given a ratings matrix in which each entry (,) represents the rating of movie by customer , if customer has watched movie and is otherwise missing, we would like to predict the remaining entries in order ...
Relaxation methods are used to solve the linear equations resulting from a discretization of the differential equation, for example by finite differences. [ 2 ] [ 3 ] [ 4 ] Iterative relaxation of solutions is commonly dubbed smoothing because with certain equations, such as Laplace's equation , it resembles repeated application of a local ...
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).