In Nielsen and Chuang's book: Quantum computation and quantum information (2016), there is an example in Box 5.4 which shows how to factor $15$ using Shor's algorithm. I am confused about a particular point in this example. They start with the state $|0\rangle|0\rangle$ and create the state $$ \frac{1}{\sqrt{2^t}}\sum_{k=0}^{2^t-1}|k\rangle|0\rangle=\frac{1}{\sqrt{2^t}}\left[|0\rangle+|1\rangle+...+|2^t-1\rangle\right]. $$ The text says they are choosing $ t=11 $ to ensure an error probability $ \epsilon $ of at most 1/4. I have been trying to figure out where this comes from. Looking back to equation $(5.35)$ it says that $$ t=n+\Big{\lceil}\log\left(2+\frac{1}{2\epsilon}\right)\Big{\rceil}. $$ For 15, $ n=4 $, right? How does $ t=11 $ work out?
Update: Okay, I have been doing alot of searching through this book and on the internet. I have come to Box 5.2 in Nielsen and Chuang (2016) and I find an expression that gives $ t=1 $. Quoting the text: "We use $ t=2L+1+\lceil\log(2+1/(2\epsilon))\rceil=\mathcal{O}(L) $, so a total of $ t-1-\mathcal{O}(L) $ squaring operations is performed at a total cost of $ \mathcal{O}(L^3) $ for the first stage." So that is where the $ t=1 $ comes from. Going back to the expression for $ t $, this works if $ n\rightarrow 2L+1 $. According to page 227 of the text (Nielsen and Chuang, 2016), this gives and estimate for the phase accurate to $ 2L+1 $ bits. Still feeling a bit confused, but I think things are becoming clear.