In reference to Hamming Code:
(n + 1) x 2^m £ 2^n
(2^m is "2 to the power m")
In context:
a code with words consisting of m data bits and r check bits
Each valid code word has n bits, and an error could occur in any of these n positions. Thus, each valid code word has n illegal code words at a distance of 1. Therefore, if we are concerned with each legal code word and each invalid code word consisting of one error, we have n + 1 bit patterns associated with each code word (1 legal word and n illegal words). Since each code word consists of n bits, where n = m + r, there are 2^n total bit patterns possible. This results in the following inequality: (n + 1) x 2^m £ 2^n where n + 1 is the number of bit patterns per code word, 2^m is the number of legal code words, and 2^n is the total number of bit patterns possible.
Thank you in advance.