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Specifically, the linear code is chosen by picking a generator matrix whose entries are randomly chosen elements of . The minimum Hamming distance of a linear code is equal to the minimum weight of a nonzero codeword, so in order to prove that the code generated by has minimum distance , it suffices to show that for any .
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 ...
The Reed–Solomon code is a [ n, k, n − k + 1] code; in other words, it is a linear block code of length n (over F) with dimension k and minimum Hamming distance The Reed–Solomon code is optimal in the sense that the minimum distance has the maximum value possible for a linear code of size ( n , k ); this is known as the Singleton bound.
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.
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 .
Richard Wesley Hamming (February 11, 1915 – January 7, 1998) was an American mathematician whose work had many implications for computer engineering and telecommunications. His contributions include the Hamming code (which makes use of a Hamming matrix ), the Hamming window, Hamming numbers, sphere-packing (or Hamming bound ), Hamming graph concepts, and the Hamming distance .
Hamming bound. In mathematics and computer science, in the field of coding theory, the Hamming bound is a limit on the parameters of an arbitrary block code: it is also known as the sphere-packing bound or the volume bound from an interpretation in terms of packing balls in the Hamming metric into the space of all possible words.
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 .