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Linear block codes are frequently denoted as [n, k, d] codes, where d refers to the code's minimum Hamming distance between any two code words. (The [n, k, d] notation should not be confused with the (n, M, d) notation used to denote a non-linear code of length n, size M (i.e., having M code words), and minimum Hamming distance d.)
The notation (,,) describes a block code over an alphabet of size , with a block length , message length , and distance . If the block code is a linear block code, then the square brackets in the notation [ n , k , d ] q {\displaystyle [n,k,d]_{q}} are used to represent that fact.
In mathematics, a block matrix or a partitioned matrix is a matrix that is interpreted as having been broken into sections called blocks or submatrices. [1] [2]Intuitively, a matrix interpreted as a block matrix can be visualized as the original matrix with a collection of horizontal and vertical lines, which break it up, or partition it, into a collection of smaller matrices.
Hence the rate of Hamming codes is R = k / n = 1 − r / (2 r − 1), which is the highest possible for codes with minimum distance of three (i.e., the minimal number of bit changes needed to go from any code word to any other code word is three) and block length 2 r − 1.
This also shows that this more abstract approach to the determinant yields the same definition as the one using the Leibniz formula. To see this it suffices to expand the determinant by multi-linearity in the columns into a (huge) linear combination of determinants of matrices in which each column is a standard basis vector. These determinants ...
In mathematics, the Kronecker product, sometimes denoted by ⊗, is an operation on two matrices of arbitrary size resulting in a block matrix.It is a specialization of the tensor product (which is denoted by the same symbol) from vectors to matrices and gives the matrix of the tensor product linear map with respect to a standard choice of basis.
Block diagram representation of the linear state-space equations. ... Due to the simplicity of this matrix notation, the state-space representation is commonly used ...
An circulant matrix takes the form = [] or the transpose of this form (by choice of notation). If each is a square matrix, then the matrix is called a block-circulant matrix.. A circulant matrix is fully specified by one vector, , which appears as the first column (or row) of .