On subfields of the Hermitian function fields involving the involution automorphism

A function field over a finite field is called maximal if it achieves the Hasse-Weil bound. Finding possible genera that maximal function fields achieve has both theoretical interest and practical applications to coding theory and other topics. As a subfield of a maximal function field is also maximal, one way to find maximal function fields is to find all subfields of a maximal function field. Due to the large automorphism group of the Hermitian function field, it is natural to find as many subfields of the Hermitian function field as possible. In literature, most of papers studied subfields fixed by subgroups of the decomposition group at one point (usually the point at infinity). This is because it becomes much more complicated to study the subfield fixed by a subgroup that is not contained in the decomposition group at one point. In this paper, we study subfields of the Hermitian function field fixed by subgroups that are not contained in the decomposition group of any point except the cyclic subgroups. It turns out that some new maximal function fields are found

Algebraic geometry codes with complementary duals exceed the asymptotic Gilbert-Varshamov bound

It was shown by Massey that linear complementary dual (LCD for short) codes are asymptotically good. In 2004, Sendrier proved that LCD codes meet the asymptotic Gilbert-Varshamov (GV for short) bound. Until now, the GV bound still remains to be the best asymptotical lower bound for LCD codes. In this paper, we show that an algebraic geometry code over a finite field of even characteristic is equivalent to an LCD code and consequently there exists a family of LCD codes that are equivalent to algebraic geometry codes and exceed the asymptotical GV bound