Standard Generator Matrix for Linear Code/Examples/(6, 3) code in Z2

Example of Standard Generator Matrix for Linear Code
Let $G$ be the standard generator matrix:


 * $G := \begin{pmatrix}

1 & 0 & 0 & 1 & 1 & 0 \\ 0 & 1 & 0 & 1 & 0 & 1 \\ 0 & 0 & 1 & 0 & 1 & 1 \end{pmatrix}$

$G$ generates the linear code $C$:


 * $C = \set {000000, 100110, 010101, 110011, 001011, 101101, 011110, 111000}$

The minimum distance of $C$ is $3$, so $C$ detects $2$ transmission errors and corrects $1$ transmission error.

Proof
Multiplying $G$ by the $8$ vectors $000, 001, 010, 011, 100, 101, 110, 111$ in turn gives:

all arithmetic being modulo $2$.

As can be seen, $2$ of the codewords have weight $3$, and $3$ have weight $4$.

From Minimum Distance of Linear Code is Smallest Weight of Non-Zero Codeword, the minimum distance of $C$ is $3$.

From Error Detection Capability of Linear Code, $C$ can detect $3 - 1 = 2$ transmission errors.

From Error Correction Capability of Linear Code, $C$ can correct $\dfrac {3 - 1} 2 = 1$ transmission error.