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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.
A generator matrix for a Reed–Muller code RM(r, m) of length N = 2 m can be constructed as follows. Let us write the set of all m-dimensional binary vectors as: = = {, …,}.
These basis codewords are often collated in the rows of a matrix G known as a generating matrix for the code C. When G has the block matrix form G = [ I k | P ] {\displaystyle {\boldsymbol {G}}=[I_{k}|P]} , where I k {\displaystyle I_{k}} denotes the k × k {\displaystyle k\times k} identity matrix and P is some k × ( n − k ) {\displaystyle ...
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:
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The generator matrix of the augmented Hadamard code is obtained by restricting the matrix to the columns whose first entry is one. For example, the generator matrix for the augmented Hadamard code of dimension = is: ′ = [].
Almost all two-dimensional bar codes such as PDF-417, MaxiCode, Datamatrix, QR Code, and Aztec Code use Reed–Solomon error correction to allow correct reading even if a portion of the bar code is damaged. When the bar code scanner cannot recognize a bar code symbol, it will treat it as an erasure.
A generator matrix for the binary Golay code is I A, where I is the 12×12 identity matrix, and A is the complement of the adjacency matrix of the icosahedron. A convenient representation. It is convenient to use the "Miracle Octad Generator" format, with co-ordinates in an array of 4 rows, 6 columns. Addition is taking the symmetric difference.
The linear independence of the columns of a generator matrix of an MDS code permits a construction of MDS codes from objects in finite projective geometry. Let (,) be the finite projective space of (geometric) dimension over the finite field .
For a systematic linear code, the generator matrix, , can always be written as = [|], where is the identity matrix of size . Examples. Checksums and hash functions, combined with the input data, can be viewed as systematic error-detecting codes.