Nonlinear complexity and weierstrass semigroup of two rational points on a Hermitian curve

O. Geil, F. Özbudak, and D. Ruano give a construction of a sequence of length (q − 1)(q2 − 1) with high nonlinear complexity by using a function on a Hermitian curve over Fq 2 with the pole divisor (q − 1)P∞ + (q − 1)Q, where P∞ is the point at infinity, Q is a rational point with the order of its orbit is q2 − 1 and q is a prime power. They give lower bounds on the k−th ordered nonlinear complexities Nk (s) and Lk (s) on the Hermitian curve. In this work, we examine the lower bounds on Nk (s) and Lk (s) using all possible pairs of rational points that can be selected on a Hermitian curve over Fq 2. In particular, we improve the bounds on Nk (s) and Lk (s) obtained by O. Geil, F. Özbudak, and D. Ruano.