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

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.

$\blacksquare$