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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.
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 ...
Generator (computer programming) For other uses, see Generator (disambiguation) § Computing. In computer science, a generator is a routine that can be used to control the iteration behaviour of a loop. All generators are also iterators. [ 1 ] A generator is very similar to a function that returns an array, in that a generator has parameters ...
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 ...
Parity-check matrix. In coding theory, a parity-check matrix of a linear block code C is a matrix which describes the linear relations that the components of a codeword must satisfy. It can be used to decide whether a particular vector is a codeword and is also used in decoding algorithms.
Python is a high-level, general-purpose programming language. Its design philosophy emphasizes code readability with the use of significant indentation. [32] Python is dynamically typed and garbage-collected. It supports multiple programming paradigms, including structured (particularly procedural), object-oriented and functional programming.
The set of codewords with Hamming weight 5 is a 3- (11,5,4) design. The generator matrix given by Golay (1949, Table 1.) is The automorphism group of the (original) ternary Golay code is the Mathieu group M11, which is the smallest of the sporadic simple groups.
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 ...