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 unique codeword of the linear code C, c is a unique row vector, and a bijection exists between w and c. A generator matrix for a ($n$, $M=q^{k}$, $d$)_$q$-code is of dimension $k*n$. 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.). Note that the number of redundant bits is denoted $r=n-k$.

The standard 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.

External links

• MathWorld entry