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 input vector. Both w and s are assumed to be row vectors. A generator matrix for a linear [ n , k , d ] q -code has format k × 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.

The standard form for a generator matrix is,

where I k 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, G = [ I k | P ] , then the parity check matrix for C is

where P ⊤ is the transpose of the matrix P . This is a consequence of the fact that a parity check matrix of C is a generator matrix of the dual code C ⊥ .

Equivalent Codes

Codes C₁ and C₂ are equivalent (denoted C₁ ~ C₂) if one code can be obtained from the other via the following two transformations:

1. arbitrarily permute the components, and
2. independently scale by a non-zero element any components.

Equivalent codes have the same minimum distance.

The generator matrices of equivalent codes can be obtained from one another via the following elementary operations:

1. permute rows
2. scale rows by a nonzero scalar
3. add rows to other rows
4. permute columns, and
5. scale columns by a nonzero scalar.

Thus, we can perform Gaussian Elimination on G. Indeed, this allows us to assume that the generator matrix is in the standard form. More precisely, for any matrix G we can find a invertible matrix U such that U G = [ I k | P ] , where G and [ I k | P ] generate equivalent codes.