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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 ...
The parity-check matrix of a Hamming code is constructed by listing all columns of length r that are non-zero, which means that the dual code of the Hamming code is the shortened Hadamard code, also known as a Simplex code. The parity-check matrix has the property that any two columns are pairwise linearly independent.
The high rank matrix completion in general is NP-Hard. However, with certain assumptions, some incomplete high rank matrix or even full rank matrix can be completed. Eriksson, Balzano and Nowak [10] have considered the problem of completing a matrix with the assumption that the columns of the matrix belong to a union of multiple low-rank subspaces.
Low-density parity-check code, also known as Gallager code, as the archetype for sparse graph codes; LT code, which is a near-optimal rateless erasure correcting code (Fountain code) m of n codes; Nordstrom-Robinson code, used in Geometry and Group Theory [31] Online code, a near-optimal rateless erasure correcting code; Polar code (coding theory)
The constituent encoders are typically accumulators and each accumulator is used to generate a parity symbol. A single copy of the original data (S 0,K-1) is transmitted with the parity bits (P) to make up the code symbols. The S bits from each constituent encoder are discarded. The parity bit may be used within another constituent code.
A qubit is a two-level system, and when we measure one qubit, we can have either 1 or 0 as a result. One corresponds to odd parity, and zero corresponds to even parity. This is what a parity check is. This idea can be generalized beyond single qubits. This can be generalized beyond a single qubit and it is useful in QEC.
A parity plot is a scatterplot that compares a set of results from a computational model against benchmark data. Each point has coordinates ( x , y ), where x is a benchmark value and y is the corresponding value from the model.
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