Hamming's Correction of Single-Bit Errors

Let's say that a single transmission consists of m bits for the message, and r bits of seemingly redundant information.
We thus transmit n = m + r  bits when transmitting a message.

The critical question is "how much additional information (the redundant bits) do we need to transmit so that the receiver can correct errors?"

Each of the 2 ^m possible message words has n illegal code words which are a distance 1 from it. Therefore each message word requires n + 1 distinct bit patterns (1 legal one, and n illegal ones).

In 1950, mathematician and Turing Award winner Richard Hamming developed a method which achieves the lower bound of:

Given a code word of 7 data bits and 4 check bits, we number the code word from 1 from the left hand side.

Each bit whose ordinal position is a power of 2 {1,2,4,8,...} is a check bit and forces the parity of some "collection" of bits including itself. Parity may be forced to be either even or odd.