Linear codes from Hermitian curves

Gabor Korchmaros
Department of Mathematics, Informatics and Economy of University of Basilicata

Pietro Speziali
Department of Mathematics, Informatics and Economy of University of Basilicata

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Minisymposium: FINITE GEOMETRY

Content: We deal with some problems in Finite geometry arising from current research on $AG$ (algebraic-geometry) codes defined over the Hermitian curve. In $PG(2,q^2)$, let $\mathcal H$ be the Hermitian curve. For a fixed positive integer $d$, let $\mathbf{S}_d$ be the family of all degree $d$ plane algebraic curves (possibly singular or reducible) defined over $\mathbf{F}_{q^2}$ which do not have $\mathcal H$ as a component. A natural question in Finite geometry is to ask for the maximum number $N(d)$ of common points in $PG(2,q^2)$ of $\mathcal H$ with $\mathcal S$ where $\mathcal S$ ranges over $\mathbf{S}_d$. From B\'ezout's theorem, $N(d)\leq d(q+1)$. The problem of finding better upper bounds on $N(d)$ for certain families of curves $\mathbf{S}$ is motivated by the ``minimum distance problem'' for $AG$-codes. A family of curves which play role in the study of Hermitian codes is defined as follows: Let $\beta$ be a Baer involution of $PG(2,q^2)$ which preserves $\mathcal H$. The set of points of $\mathcal H$ which are fixed by $\beta$ has size $q+1$ and it is the complete intersection $\mathcal H\cap\mathcal C^2$ of $\mathcal H$ with an irreducible conic $\mathcal C^2$. For a positive integer $m$ less than $q+1$, define $D$ to be the set of all points of $\mathcal H$ other than those in $U$, together with the divisor (formal sum) $\textsf{G}$ of points on $\mathcal H$: $$\textsf{G}:=m\sum_{{\mathcal H}\cap{\mathcal C^2}} P.$$ The functional $AG$-code $C_L(\textsf{G},D)$ is obtained taking the rational functions with pole numbers at most $m$ at any point in $\mathcal H\cap\mathcal C^2$ and evaluating them at the points of $D$. For $m$ even, the minimum distance problem for $C_L(D,G)$ is equivalent to the problem of determining the maximum number of points on $\mathcal H\cap \mathcal S$ with $\mathcal S\in \mathbf{S}_m$. For $m$ odd, let $\mathbf{T}_{m+1}$ be the subfamily of $\mathbf{S}_d$ consisting of all curves $\mathcal T$ of degree $m+1$ which contain all points in $D$. The minimum distance problem for $C_L(D,G)$ is equivalent to the problem of determining the maximum number of points on $\mathcal H\cap \mathcal T$ with $\mathcal T\in \mathbf{T}_{m+1}$. The automorphism group of $C_L(D,G)$ contains a subgroup $G\cong P\Gamma L(2,q)$. Is it true that this subgroup is the full automorphism group? We address the above problems using tools from Finite geometry.

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