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

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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