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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.
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 . In particular, an equivalent way to write the inner product definition for the Hadamard code arises by using the generator ...
Reed–Muller codes are linear block codes that are locally testable, locally decodable, and list decodable. These properties make them particularly useful in the design of probabilistically checkable proofs. Traditional Reed–Muller codes are binary codes, which means that messages and codewords are binary strings.
Hamming codes can be computed in linear algebra terms through matrices because Hamming codes are linear codes. For the purposes of Hamming codes, two Hamming matrices can be defined: the code generator matrix G and the parity-check matrix H: Bit position of the data and parity bits As mentioned above, rows 1, 2, and 4 of G should look familiar as they map the data bits to their parity bits: p1 ...
The proof uses the probabilistic method. High-level proof: To show the existence of the linear code that satisfies those constraints, the probabilistic method is used to construct the random linear code. Specifically, the linear code is chosen by picking a generator matrix whose entries are randomly chosen elements of .
The linear independence of the columns of a generator matrix of an MDS code permits a construction of MDS codes from objects in finite projective geometry. Let be the finite projective space of (geometric) dimension over the finite field .
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.
Roblox (/ ˈroʊblɒks / ROH-bloks) is an online game platform and game creation system developed by Roblox Corporation that allows users to program and play games created by themselves or other users. Created by David Baszucki and Erik Cassel in 2004 and released in 2006, the platform hosts user-created games of multiple genres coded in the ...