Generator matrix

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 ($n$, $M=q^{k}$, $d$)_$q$-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

$G={\begin{bmatrix}I_{k}|P\end{bmatrix}}$

where $I_{k}$ 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).

Equivalent Codes

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

1. permute components, and
2. scale components.

Equivalent codes have the same distance.

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

1. permute rows
2. scale rows
3. add rows
4. permute columns, and
5. scale columns.

See also

Hamming code (7,4)

External links

• MathWorld entry