Generator matrix

In coding theory , a generator matrix , also known as a generator matrix , is a matrix-like basis for a linear code that generates all possible code words. If G is a generator matrix for a linear [n, k] code C then every code word c of C is of the form

${\displaystyle c=wG}$

for a unique row vector w with k entries. In other words: the image is a bijection. A generator matrix for a code has the format . Here n is the length of the code words and k is the number of information bits (the dimension of C ). The number of redundant bits is r = n - k . ${\displaystyle K^{1\times k}\rightarrow C,w\mapsto wG}$ ${\displaystyle [n,k]}$ ${\displaystyle C}$ ${\displaystyle k\times n}$

The systematic form for a generator matrix is

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

where a k × k identity matrix and P is of dimension k × r . ${\displaystyle I_{k}}$

A generator matrix can be used to generate a control matrix for a code (and vice versa).

Equivalent codes

Codes C ₁ and C ₂ are equivalent (written C ₁ ~ C ₂ ) if one code can be generated from the other by the following two transformations

1. Swap components
2. Scale components.

Equivalent codes have the same Hamming distance.

The generator matrices of equivalent codes can be generated using the following transformations:

1. Swap lines
2. Scale lines
3. Add lines
4. Swap columns
5. Scale columns.

See also

• Hamming code

Web links

• MathWorld entry (English)