Linear feedback shift registers (LFSR's) work by taking a fixed-length bit-string $b\in\{0,1\}^n$, as well as fixed "taps" (bit positions) and applying XOR to the taps, giving one output bit, which is appended at the $b$ after shifting it.
Now XOR is a linear function. A natural non-linear function that can be used on the fixed set of taps is a kind of "majority vote", which works as follows: if the majority of the taps is $0$, then we output $1$, and vice versa. (For this, it is best to have an odd number of taps.)
A simple implementation of a majority-based feedback shift register can be found here.
Of course, applying this "majority vote" procedure over and over again, this eventually gets periodical.
Question. Given fixed bit-length $n$, what is a lower bound of the maximal length of a period that can be achieved choosing a suitable start string $b\in \{0,1\}^n$ and a suitable set of taps?
Also, I wasn't able to find out if and where this concept has been studied.