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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.
A generator matrix for a Reed–Muller code RM (r, m) of length N = 2m can be constructed as follows. Let us write the set of all m -dimensional binary vectors as:
The core of the full Aztec code. 40 bits are available between the orientation marks for encoding parameters. Message data is placed in a spiral pattern around the core. The mode message begins "01011100", indicating 01 2 +1 = 2 layers, and 011100 2 +1 = 29 data codewords (of 6 bits each). The compact Aztec code core may be surrounded by 1 to 4 ...
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] That is, they show how linear combinations of ...
In linear algebra terms, the dual code is the annihilator of C with respect to the bilinear form . The dimension of C and its dual always add up to the length n: A generator matrix for the dual code is the parity-check matrix for the original code and vice versa. The dual of the dual code is always the original code.
Basic general information about the generators, including: creator or company, license, and price.
A UPC-A barcode. A barcode or bar code is a method of representing data in a visual, machine-readable form.Initially, barcodes represented data by varying the widths, spacings and sizes of parallel lines.
The Hadamard code is a linear code, and all linear codes can be generated by a generator matrix . This is a matrix such that holds for all , where the message is viewed as a row vector and the vector-matrix product is understood in the vector space over the finite field .