Bulletin of the
Korean Mathematical Society BKMS

{O. Geil, F. "Ozbudak, and D. Ruano give a construction of a sequence of length $(q-1)(q^{2}-1)$ with high nonlinear complexity by using a function on a Hermitian curve over $F_{q^{2}}$ with the pole divisor $ (q-1)P_{infty}+(q-1)Q,$ where $ P_{infty}$ is the point at infinity, $Q$ is a rational point with the order of its orbit is $q^2-1$ and $q$ is a prime power. They give lower bounds on the $k-$th ordered nonlinear complexities $N^{k}(s)$ and $L^{k}(s)$ on the Hermitian curve. In this work, we examine the lower bounds on $N^{k}(s)$ and $L^{k}(s)$ using all possible pairs of rational points that can be selected on a Hermitian curve over $F_{q^{2}}.$ In particular, we improve the bounds on $N^{k}(s)$ and $L^{k}(s)$ obtained by O. Geil, F. "Ozbudak, and D. Ruano.}

Keywords: Weierstrass semigroup, Hermitian function field, nonlinear complexity

MSC numbers: Primary 94B27,11T71,14H55