Why parity bits sit at positions 1, 2, 4, 8 — Hamming code

Why this example is worth doing

Each parity bit at position 2ᵏ covers exactly those positions whose k-th bit is set, so every data position is covered by a unique combination of parity bits — its own binary expansion. The syndrome therefore cannot help but spell out the error position. The page presents the coverage table and lets the reader verify that no two positions share a pattern, which is the proof that single errors are always locatable.

Try your own input in the Hamming code. Encode, inject an error, and watch the syndrome point straight at the flipped bit.