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"=cGwhere "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: 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

* [http://mathworld.wolfram.com/GeneratorMatrix.html MathWorld entry]