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 ${\displaystyle r=n-k}$.

The standard form for a generator matrix is,

where ${\displaystyle I_{k}}$ is the ${\displaystyle k\times k}$ identity matrix and P is a ${\displaystyle k\times (n-k)}$ matrix. When the generator matrix is in standard form, the code C is systematic in its first k coordinate positions.

A generator matrix can be used to construct the parity check matrix for a code (and vice versa). If the generator matrix G is in standard form, ${\displaystyle G={\begin{bmatrix}I_{k}|P\end{bmatrix}}}$, then the parity check matrix for C is

where ${\displaystyle P^{\top }}$ is the transpose of the matrix ${\displaystyle P}$. This is a consequence of the fact that a parity check matrix of ${\displaystyle C}$ is a generator matrix of the dual code ${\displaystyle C^{\perp }}$.

G is a ${\displaystyle k\times n}$ matrix, while H is a ${\displaystyle (n-k)\times n}$ matrix.