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The generator matrix The Reed–Muller RM (r, m) code of order r and length N = 2 m is the code generated by v0 and the wedge products of up to r of the vi, 1 ≤ i ≤ m (where by convention a wedge product of fewer than one vector is the identity for the operation).
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 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 .
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.
Linear code In coding theory, a linear code is an error-correcting code for which any linear combination of codewords is also a codeword. Linear codes are traditionally partitioned into block codes and convolutional codes, although turbo codes can be seen as a hybrid of these two types. [1] Linear codes allow for more efficient encoding and decoding algorithms than other codes (cf. syndrome ...
Days after reportedly calling Milwaukee “horrible” behind closed doors, former President Donald Trump will rally with his supporters just south of the city in one of the country’s top ...
LDPC codes are also known as Gallager codes, in honor of Robert G. Gallager, who developed the LDPC concept in his doctoral dissertation at the Massachusetts Institute of Technology in 1960. [4] [5] However, LDPC codes require computationally expensive iterative decoding, so they went unused for decades. In 1993 the newly invented turbo codes demonstrated that codes with iterative decoding ...