Construction of sequences with high Nonlinear Complexity from the Hermitian Function Field

We provide a sequence with high nonlinear complexity from the Hermitian function field $\mathcal{H}$ over $\mathbb{F}_{q^2}$. This sequence was obtained using a rational function with pole divisor in certain $\ell$ collinear rational places on $\mathcal{H}$, where $2 \leq \ell \leq q$. In particular we improve the lower bounds on the $k$th-order nonlinear complexity obtained by H. Niederreiter and C. Xing; and O. Geil, F. \"Ozbudak and D. Ruano

Complete set of Pure Gaps in Function Fields

In this work, we provide a way to completely determine the set of pure gaps $G_0(P_1, P_2)$ at two rational places $P_1, P_2$ in a function field $F$ over a finite field $\mathbb{F}_q$, and its cardinality. Furthermore, we given a bound for the cardinality of the set $G_0(P_1, P_2)$ which is better, in some cases, than the generic bound given by Homma and Kim. As a consequence, we completely determine the set of pure gaps and its cardinality for two families of function fields: the $GK$ function field and Kummer extensions.Comment: 22 page

The Set of Pure Gaps at Several Rational Places in Function Fields

In this work, using maximal elements in generalized Weierstrass semigroups and its relationship with pure gaps, we extend the results in \cite{CMT2024} and provide a way to completely determine the set of pure gaps at several rational places in an arbitrary function field $F$ over a finite field and its cardinality. As an example, we determine the cardinality and a simple explicit description of the set of pure gaps at several rational places distinct to the infinity place on Kummer extensions, which is a different characterization from that presented by Hu and Yang in \cite{HY2018}. Furthermore, we present some applications in coding theory and AG codes with good parameters