My question is, which is better - $H_3$ Hamming code, or $\hat{H_3}$ extended Hamming code. The reason that I ask is that the $P_{err}$ for the two are equal, so the chance of error between the two is the same. I am unsure why one would use the extended code for this reason.
One reason I was thinking, was the extended code is a $[8,4,4]$ code, verses a $[7,4,3]$ code - so the extended can correct an additional error. However, at the same time the extended codes are of length $8$, so that extra error correction seems to be redundant.
Thanks for any help in being able to understand which would be better.
The questions "which code is better?" cannot be answered, in general, unless you specify a goodness criterion. In regards to error correction/detection, for a forward error correcting code (ie. with no feedback) there are several measures:
1 Up to how many error bits can it correct?
2 Up to how many error bits can it detect?
3 How many errors bits in a burst can it detect?
4 What is the the probability of erroneous decoding $p(err)$? (assuming some statistical model for the channel, usually BSC - and some decoding strategy -usually hard ML)
Whichever goodeness measure you choose among these ones (or any other), it should also be combined with other factors: mainly code rate and decoding complexity.
The extended Hamming Code $(8,4)$ can be preferable to the classical Hamming Code $(7,4)$ if we are interested in error detection (criterion 2): it can detect up to 3 erroneous bits - and even then, the advantage should be weighted against the disadvantage (lower rate). It is not preferable if we are interested in error correction (criterion 1).