I've always been thought that the fastest way to sort an array of numbers has complexity $O(n \log (n))$. However, radix sort has complexity $O(kn)$ where $k$ is the number of bits. There are even questions on the internet where it is asked to prove that a sorting algorithm cannot be faster than $n \log (n)$.
Wanted to have a clarification on this. Does radix sort have any limitations? If not, is the lower bound on sorting linear in the number of elements?
The number of bits $k$ cannot be considered constant in general. In fact, if all the $n$ numbers are distinct then $k = \mathcal{\Omega}(\log n)$. Hence, there is no difference between radix sort and other fast sorting algorithms.
More generally, any generic deterministic sorting algorithm cannot better $\mathcal{O}(n \log n)$ complexity. If you have $n$ numbers, then the number of comparisons you need to make is at least $\log_2(n!)$.