In coding theory, a generator matrix is a basis for a linear code, generating all its possible codewords. If the matrix is G and the linear code is C,
- w = cG
where w is a codeword of the linear code C, c is a row vector, and a bijection exists between w and c. A generator matrix for an (, )-code has dimensions k×n. Here n is the length of a codeword, k is the number of information bits, d is the minimum distance of the code, and q 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 systematic form for a generator matrix is
where is a k×k identity matrix and P is of dimension k×r.
A generator matrix can be used to construct the parity check matrix for a code (and vice-versa).
Read more about Generator Matrix: Equivalent Codes
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