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A generator matrix for a linear -code has format , where n is the length of a codeword, k is the number of information bits (the dimension of C as a vector subspace), d is the minimum distance of the code, and q is size of the finite field, that is, the number of symbols in the alphabet (thus, q = 2 indicates a binary code, etc.).
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.
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.
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 :
Reed–Solomon codes are a group of error-correcting codes that were introduced by Irving S. Reed and Gustave Solomon in 1960. [1] They have many applications, including consumer technologies such as MiniDiscs, CDs, DVDs, Blu-ray discs, QR codes, Data Matrix, data transmission technologies such as DSL and WiMAX, broadcast systems such as satellite communications, DVB and ATSC, and storage ...
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 ).
Traditional Reed–Muller codes are binary codes, which means that messages and codewords are binary strings. When r and m are integers with 0 ≤ r ≤ m, the Reed–Muller code with parameters r and m is denoted as RM ( r , m ). When asked to encode a message consisting of k bits, where holds, the RM ( r , m) code produces a codeword consisting of 2 m bits.
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 .