Comparison of Hamming Codes

Considering (7,4) and (15, 11) Hamming codes.

Is there an intuitive idea why (15,11) performs better than (7, 4) in AWGN?

Is there anything to do with code rate?

The following curves show BER (bit error rate) versus Eb/No (energy of bit per noise density) of AWGN channel. It is possible to see that when bit has more energy (for fixed No), Hamming (15,11) performs better than (7,4), but the two codes can correct 1 bit. • Please explain all symbols, like BER, $E_b$, $N_0$, and what your performance criterium is. – Pieter21 Nov 15 '16 at 12:22
• @Pieter21, edited the question, is it better? – Bruno A Nov 15 '16 at 13:34
• This is a bit complicated. Code rate has quite a bit to do with it. One of your codes can correct one error per every seven bits, the other only one error per 15 bits. On the other hand if you normalize it per payload bit, then the higher rate code can spend more energy per bit for the same total energy consumption. Also, the variance of the number of errors per code block often make longer codes of the same rate work better. Furthermore, it may depend on whether you do hard decision or soft decision decoding. – Jyrki Lahtonen Nov 15 '16 at 13:54
• I won't give even an educated guess of how that plot was created, because there is an awful lot that you need to describe in detail. I simply refer you to an old MO answer of mine, where I discuss how to do some calculations for comparing two alternative codes. But that is hard decision decoding only, and not fully relevant. My go-to-answer is that you write soft decision decoding algorithms (easy for these codes, because basically you only need to implement Fast Walsh-Hadamard transformation), and then run a simulation. – Jyrki Lahtonen Nov 15 '16 at 13:57
• Why that question was considered on-topic at MathOverflow? Beats me :-/ – Jyrki Lahtonen Nov 15 '16 at 14:03