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A generator matrix for a Reed–Muller code RM (r, m) of length N = 2m can be constructed as follows. Let us write the set of all m -dimensional binary vectors as:
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.
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 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 .
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 :
LDPC codes have been shown to have ideal combinatorial properties. In his dissertation, Gallager showed that LDPC codes achieve the Gilbert–Varshamov bound for linear codes over binary fields with high probability. In 2020 it was shown that Gallager's LDPC codes achieve list decoding capacity and also achieve the Gilbert–Varshamov bound for linear codes over general fields. [9]
Convolutional codes are often characterized by the base code rate and the depth (or memory) of the encoder . The base code rate is typically given as , where n is the raw input data rate and k is the data rate of output channel encoded stream. n is less than k because channel coding inserts redundancy in the input bits.
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 ...