### Higher order Hamming codes

A complete family of Hamming codes exist, some of which are shown here. As the code length increases, the codes give similar correction performance at a given BER, but with ever-reducing coding overhead. In addition, the asymptotic coding gain increases with increasing code length. For example, the (4,7) Hamming code has an asymptotic coding gain of about 0.5 dB while the (120,127) Hamming code has a coding gain in the order of 1.5 dB. The code rate is higher for the (120,127) code, R = 120/127, than the (4,7) code, R = 4/7, and so has less redundancy.