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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.
This property can be easily shown based on the idea of defining a generator matrix for the concatenated code in terms of the generator matrices of Cout and Cin .
The BCH code with and higher has the generator polynomial. This code has minimal Hamming distance 15 and corrects 7 errors. It has 1 data bit and 14 checksum bits. It is also denoted as: (15, 1) BCH code. In fact, this code has only two codewords: 000000000000000 and 111111111111111 (a trivial repetition code ).
In the linear code case a different proof of the Singleton bound can be obtained by observing that rank of the parity check matrix is . [4] Another simple proof follows from observing that the rows of any generator matrix in standard form have weight at most .
Reed–Muller codes generalize the Reed–Solomon codes and the Walsh–Hadamard code. 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 .
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 .
The matrix product of the generator and parity-check matrices, , is the matrix of all zeroes, and by intent. Indeed, this is an example of the very definition of any parity check matrix with respect to its generator matrix.
In mathematics and electronics engineering, a binary Golay code is a type of linear error-correcting code used in digital communications. The binary Golay code, along with the ternary Golay code, has a particularly deep and interesting connection to the theory of finite sporadic groups in mathematics. [1] These codes are named in honor of Marcel J. E. Golay whose 1949 paper [2] introducing ...