Introduction to finite fields and their applications: Preface

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Algebraic Geometric Codes

In this article we construct for any prime power $q$ and odd $n \ge 5$, a new $\mathbb{F}_{q^{2n}}$-maximal curve $\mathcal X_n$. Like the Garcia--G\" uneri--Stichtenoth maximal curves, our curves generalize the Giulietti--Korchm\'aros maximal curve, though in a different way. We compute the full automorphism group of $\mathcal X_n$, yielding that it has precisely $q(q^2-1)(q^n+1)$ automorphisms. Further, we show that unless $q=2$, the curve $\mathcal{X}_n$ is not a Galois subcover of the Hermitian curve. Finally, we find new values of the genus spectrum of $\mathbb{F}_{q^{2n}}$-maximal curves, by considering some Galois subcovers of $\mathcal X_n$.

A new family of maximal curves

Complete permutation polynomials from exceptional polynomials

In this paper, algebraic-geometric (AG) codes associated with the GGS maximal curve are investigated. The Weierstrass semigroup at all $\mathbb F_{q^2}$-rational points of the curve is determined; the Feng-Rao designed minimum distance is computed for infinite families of such codes, as well as the automorphism group. As a result, some linear codes with better relative parameters with respect to one-point Hermitian codes are discovered. Classes of quantum and convolutional codes are provided relying on the constructed AG codes.

AG codes and AG quantum codes from the GGS curve

Some Ree and Suzuki curves are not Galois covered by the Hermitian curve

In this paper we investigate multi-point Algebraic–Geometric codes associated to the GK maximal curve, starting from a divisor which is invariant under a large automorphism group of the curve. We construct families of codes with large automorphism groups.

Giovanni Zini

Papers

Complete permutation polynomials from exceptional polynomials

AG codes and AG quantum codes from the GGS curve

Some Ree and Suzuki curves are not Galois covered by the Hermitian curve

Multi point AG codes on the GK maximal curve

On monomial complete permutation polynomials