Bonus Generalize Hamming 7 4 code to a code with r check bit

Bonus: Generalize Hamming [7, 4] code to a code with r check bits.

Solution

Hamming codes are a family of linear error-correcting codes that generalize the Hamming(7,4)-code, and were invented by Richard Hamming in 1950. Hamming codes can detect up to two-bit errors or correct one-bit errors without detection of uncorrected errors. By contrast, the simple parity code cannot correct errors, and can detect only an odd number of bits in error. Hamming codes are perfect codes, that is, they achieve the highest possible rate for codes with their block length and minimum distance of three.

Hamming codes are a class of binary linear codes. For each integer r 2 there is a code with block length n = 2^r 1 and message length k = 2^r r 1. Hence the rate of Hamming codes is R = k / n

= 1 r / (2^r 1), which is the highest possible for codes with minimum distance of three (i.e., the minimal number of bit changes needed to go from any code word to any other code word is three) and block length 2^r 1. The parity-check matrix of a Hamming code is constructed by listing all columns of length r that are non-zero, which means that the dual code of the Hamming code is the shortened Hadamard code. The parity-check matrix has the property that any two columns are pairwise linearly independent.