Counting unitary transformations in $SU(N)$

I'm referring to the following article 1, in particular to section 6.

The goal is to estimate the number of unitary transformations in $$SU(N)$$, identifying unitaries within balls of radius $$\epsilon$$. The strategy is to take the total volume of $$SU(N)$$ (see also 2), and then dividing for the volume of an $$\epsilon$$-ball. Here and in the following $$N=2^K$$, where $$K$$ is an integer.

The total volume of $$SU(N)$$ is: $$$$\frac{2 \pi^{\frac{(N+2)(N-1)}{2}}}{1! 2! 3!\cdots(N-1)!}$$$$
Here the author misses a total factor of $$\sqrt{N 2^{N-3}}$$, see the original article 2, equation 5.13. Despite this, the author then states that the volume of an $$\epsilon$$-ball of dimension $$N^2-1$$ is: $$$$\frac{\pi^{\frac{N^2-1}{2}}}{\left(\frac{N^2-1}{2}\right)!}$$$$
I don't get this last formula. First of all, I believe that there would be a factor $$\epsilon$$ to some power of $$N$$. Secondly, it seems to me the volume of an $$\epsilon$$-ball in an Euclidean space off even dimension, while considering that $$N=2^K$$, $$N^2-1$$ is odd. Thirdly, in my opinion it would be better to consider the volume of an $$\epsilon$$-ball in $$SU(N)$$ (if I didn't misunderstood the formula above).

My questions are:

• Has someone, reading the article 1, reached a better comprehension than mine on the above formulas?

• If no, may you have other references about estimating the number of unitary transformations in $$SU(N)$$ within a precision $$\epsilon$$?

• And, finally, has someone a reference for the volume of an $$\epsilon$$-ball in $$SU(N)$$? Looking on the net I didn't find anything until now.

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