The nonlinearity and Hamming weights of rotation symmetric Boolean functions of small degree

Let $ e $, $ l $ and $ n $ be integers such that $ 1\le e \lt n $ and $ 3\le l\le n $. Let $ \left\langle{i}\right\rangle $ denote the least nonnegative residue of $ i \mod n $. In this paper, we investigate the following Boolean function $ F_{l, e}^n(x^n) = \sum\limits_{i = 0}^{n-1}x_{i} x_{\left\langle{i+e}\right\rangle}x_{\left\langle{i+2e}\right\rangle}...x_{\left\langle{i+(l-1)e}\right\rangle}, $ which plays an important role in cryptography and coding theory. We introduce some new sub-functions and provide some recursive formulas for the Fourier transform. Using these recursive formulas, we show that the nonlinearity of $ F_{l, e}^n(x^n) $ is the same as its weight for $ 5\leq l\leq 7 $. Our result confirms partially a conjecture of Yang, Wu and Hong raised in 2013. It also gives a partial answer to a conjecture of Castro, Medina and Stănică proposed in 2018. Our result extends the result of Zhang, Guo, Feng and Li for the case $ l = 3 $ and that of Yang, Wu and Hong for the case $ l = 4 $.