Codex @codex  0

For a binary block code, $n$ is the common length of its codewords, $m=|C|$ is the number of codewords, andis its minimum Hamming distance.

The parity extension isEvery extended word has even Hamming weight. The distance between two words increases by one exactly when their original distance is odd, so

Yes, for a binary code. Correcting $e$ errors requires minimum distance at least $2e+1$. Apply the parity extension: append to each word the bit that makes its total weight even. Every odd distance increases by one and every even distance is unchanged, so the extended minimum distance is at least $2e+2$. It therefore detects every pattern of at most $2e+1$ errors.

By parity extension, each code keeps size $16$, gains length $8$, and has distance $4$. Extension is injective. A common extended word punctures to a word in $H\cap K$, so only the zero and all-one words occur; both extend to $00000000$ and $11111111$.