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Generator matrix. In coding theory, a generator matrix is a matrix whose rows form a basis for a linear code. The codewords are all of the linear combinations of the rows of this matrix, that is, the linear code is the row space of its generator matrix.
For general , the generator matrix of the augmented Hadamard code is a parity-check matrix for the extended Hamming code of length and dimension , which makes the augmented Hadamard code the dual code of the extended Hamming code. Hence an alternative way to define the Hadamard code is in terms of its parity-check matrix: the parity-check ...
The generator matrix [ edit ] The Reed–Muller RM( r , m ) code of order r and length N = 2 m is the code generated by v 0 and the wedge products of up to r of the v i , 1 ≤ i ≤ m (where by convention a wedge product of fewer than one vector is the identity for the operation).
This matrix is a Vandermonde matrix over . In other words, the Reed–Solomon code is a linear code , and in the classical encoding procedure, its generator matrix is A {\displaystyle A} . Systematic encoding procedure: The message as an initial sequence of values
MATLAB (an abbreviation of "MATrix LABoratory" [22]) is a proprietary multi-paradigm programming language and numeric computing environment developed by MathWorks. MATLAB allows matrix manipulations, plotting of functions and data, implementation of algorithms , creation of user interfaces , and interfacing with programs written in other languages.
Permutation matrix. In mathematics, particularly in matrix theory, a permutation matrix is a square binary matrix that has exactly one entry of 1 in each row and each column with all other entries 0. [1]: 26 An n × n permutation matrix can represent a permutation of n elements. Pre- multiplying an n -row matrix M by a permutation matrix P ...
The second row is the same generator with a seed of 3, which produces a cycle of length 2. Using a = 4 and c = 1 (bottom row) gives a cycle length of 9 with any seed in [0, 8]. A linear congruential generator (LCG) is an algorithm that yields a sequence of pseudo-randomized numbers calculated with a discontinuous piecewise linear equation.
The fact that the Pauli matrices, along with the identity matrix I, form an orthogonal basis for the Hilbert space of all 2 × 2 complex Hermitian matrices means that we can express any Hermitian matrix M as = + where c is a complex number, and a is a 3-component, complex vector.