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:

\(\ds 1 + 0\)

\(=\)

\(\ds 1\)

\(\ds \)

\(\equiv\)

\(\ds 1\)

\(\ds \pmod 2\)

\(\ds 1 + 1\)

\(=\)

\(\ds 2\)

\(\ds \)

\(\equiv\)

\(\ds 0\)

\(\ds \pmod 2\)

\(\ds 0 + 1\)

\(=\)

\(\ds 1\)

\(\ds \)

\(\equiv\)

\(\ds 1\)

\(\ds \pmod 2\)

Hence the result.

$\blacksquare$

Sources

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