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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.
Either of degree 11 irreducible factors can be used to generate the code. [6] Turyn's construction of 1967, "A Simple Construction of the Binary Golay Code," that starts from the Hamming code of length 8 and does not use the quadratic residues mod 23. [7] From the Steiner System S(5,8,24), consisting of 759 subsets of a 24-set. If one ...
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).
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]
LDPC codes functionally are defined by a sparse parity-check matrix. This sparse matrix is often randomly generated, subject to the sparsity constraints—LDPC code construction is discussed later. These codes were first designed by Robert Gallager in 1960. [5] Below is a graph fragment of an example LDPC code using Forney's factor graph notation.
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.
Definition. Fix a finite field ... The generator polynomial of the BCH code is defined as the least common multiple = ((), ... In matrix form, we have ...
Linear combinations, or vector addition, of the rows of the matrix produces all possible words contained in the code. This is referred to as the span of the rows. The inner product of any two rows of the generator matrix will always sum to zero. These rows, or vectors, are said to be orthogonal.