Which is better - $H_3$ Hamming code, or $\hat{H_3}$ extended Hamming code. 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.
 A: 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). 
