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.

Terminology

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 vector. A generator matrix for a linear -code has format , 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.

The standard form for a generator matrix is,

where is the k×k identity matrix and P is a k×r 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, , then the parity check matrix for C is