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We can obtain a Walsh matrix from a Hadamard matrix. For that, we first generate the Hadamard matrix for a given dimension. Then, we count the number of sign changes of each row. Finally, we re-order the rows of the matrix according to the number of sign changes in ascending order.
Least squares is part of a broader class of regression-based methods lumped together here for simplicity. These regression formulae minimize the residual differences between path-distances along the tree and pairwise distances in the data matrix, effectively "fitting" the tree to the empirical distances.
Compared to randomly generated LDPC codes, structured LDPC codes—such as the LDPC code used in the DVB-S2 standard—can have simpler and therefore lower-cost hardware—in particular, codes constructed such that the H matrix is a circulant matrix. [30] Yet another way of constructing LDPC codes is to use finite geometries.
The code generally known simply as the ternary Golay code is an [,,]-code, that is, it is a linear code over a ternary alphabet; the relative distance of the code is as large as it possibly can be for a ternary code, and hence, the ternary Golay code is a perfect code.
In finite geometry, the Fano plane (after Gino Fano) is a finite projective plane with the smallest possible number of points and lines: 7 points and 7 lines, with 3 points on every line and 3 lines through every point.
Linear codes are usually implemented as systematic error-correcting codes (e.g., Reed-Solomon codes in CDs). Convolutional codes are implemented as either systematic or non-systematic codes. Non-systematic convolutional codes can provide better performance under maximum-likelihood decoding.
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]
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 ...