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.

If G is a matrix, it generates the codewords of a linear code C by,

where w is a codeword of the linear code C, and s is any input vector. Both w and s are assumed to be row vectors. A generator matrix for a linear ${\displaystyle [n,k,d]_{q}}$-code has format ${\displaystyle k\times n}$, 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.). The number of redundant bits is denoted by r = n - k.

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