# Addition of Codewords in Linear Code/Examples/V(3,2)

Jump to navigation
Jump to search

## Example of Addition of Codewords in $V \paren {3, 2}$

In the master code $V \paren {3, 2}$, the codewords $110$ and $011$ are added thus:

- $110 + 011 = 101$

## Proof

Taking the elements of $110$ and $011$ in pairs, successively, we have:

\(\displaystyle 1 + 0\) | \(=\) | \(\displaystyle 1\) | |||||||||||

\(\displaystyle \) | \(\equiv\) | \(\displaystyle 1\) | \(\displaystyle \pmod 2\) |

\(\displaystyle 1 + 1\) | \(=\) | \(\displaystyle 2\) | |||||||||||

\(\displaystyle \) | \(\equiv\) | \(\displaystyle 0\) | \(\displaystyle \pmod 2\) |

\(\displaystyle 0 + 1\) | \(=\) | \(\displaystyle 1\) | |||||||||||

\(\displaystyle \) | \(\equiv\) | \(\displaystyle 1\) | \(\displaystyle \pmod 2\) |

Hence the result.

$\blacksquare$

## Sources

- 1996: John F. Humphreys:
*A Course in Group Theory*... (previous) ... (next): Chapter $6$: Error-correcting codes