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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.
Definition. Formally, a parity check matrix H of a linear code C is a generator matrix of the dual code, C⊥. This means that a codeword c is in C if and only if the matrix-vector product Hc⊤ = 0 (some authors [1] would write this in an equivalent form, cH⊤ = 0 .)
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: X = F 2 m = { x 1 , … , x N } . {\displaystyle X=\mathbb {F} _{2}^{m}=\{x_{1},\ldots ,x_{N}\}.}
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: ′ = [].
If C is a code with a generating matrix G in standard form, = [|], then = [|] is a check matrix for C. The code generated by H is called the dual code of C. It can be verified that G is a k × n {\displaystyle k\times n} matrix, while H is a ( n − k ) × n {\displaystyle (n-k)\times n} matrix.
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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 :
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.
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. Self-dual codes. A self-dual code is one which is its own dual. This implies that n is even and dim C = n/2.
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.
Used in a football pool with 11 games, the ternary Golay code corresponds to 729 bets and guarantees exactly one bet with at most 2 wrong outcomes. The set of codewords with Hamming weight 5 is a 3- (11,5,4) design . The generator matrix given by Golay (1949, Table 1.) is.