\section{Introduction}
\label{intr}
Let $q$ be a prime power and let $\numberset{P}^2$ denote the projective plane over the
field $\numberset{F}_{q^2}$. Let $X \subseteq \numberset{P}^2$ be the Hermitian curve (see
\cite{Sti}, Example VI.3.6) of affine equation $y^q+y=x^{q+1}$. It is well-known
that $X$ is a maximal curve with $q^3+1$ $\numberset{F}_{q^2}$-rational points (for
instance, \cite{Ste}). Choose $m,n \in \numberset{Z}$ and two distinct points $P,Q \in
X(\numberset{F}_{q^2})$. The code obtained evaluating the vector space $L(mP+nQ)$ on
$X(\numberset{F}_{q^2})\setminus \{ P,Q\}$ is said to be a (classical) two-point code on
$X$ (\cite{HK1}, Section 1). Obviously we may consider only the pairs $(m,n)$
such that $n+m>0$. Otherwise the code is not of interest. Since the group of
automorphisms of $X$, say $\mbox{Aut}(X)$, is 2-transitive (\cite{AC}, pages
572-573) we can assume, without loss of generality, that $P=P_\infty$ and
$Q=P_0$, where $P_\infty$ is the  only point at infinity of $X(\numberset{F}_{q^2})$, of
projective coordinates $(0:1:0)$, and $P_0$ is the point of coordinates $(0:0:1)
\in X(\numberset{F}_{q^2})$. Set $B:=X(\numberset{F}_{q^2}) \setminus \{ P_\infty,P_0\}$ and denote by
$C_{m,n}$ the two-point code obtained evaluating the vector space
$L(mP_\infty+nP_0)$ on $B$.
The minimum distance of such codes has been completely determined by Homma and
Kim in \cite{HK1} and \cite{HK2}. Using different techniques, Park gave in
\cite{Park} explicit formulas for their dual minimum distance. We will widely
use the following two results\footnote{If $X$ is the Hermitian curve and $P \in
X(\numberset{F}_{q^2})$ then any canonical $\numberset{F}_{q^2}$-rational divisor $K$ on $X$ satisfies
$K \sim (q-2)(q+1)P$.}.
\begin{theorem}[\cite{Park}, Theorem 3.3]
\label{3.3}
Let $G=K+mP_\infty + nP_0$, where $K$ is a canonical divisor on $X$. Let
$C^\perp$ be the dual of the code obtained evaluating the vector space $L(G)$ on
the points of $B$ and denote by $\delta$ its minimum distance. Write
\begin{eqnarray*}
m&=&m_0(q+1)-m_1, \ \ \  0 \le m_1 \le q,\\
n&=&n_0(q+1)-n_1, \ \ \ \ 0 \le n_1 \le q. 
\end{eqnarray*}
Set $d^*:=m+n$ and suppose that $G$ satisfies either
\begin{itemize}
\item[(a)] $\deg(G)>\deg(K)+q$, or
\item[(b)] $\deg(K) \le \deg(G) \le \deg(K)+q$ and $G \not \sim sP_\infty$ and
$G \not \sim tP_0$, for all $s,t \in \numberset{Z}$.
\end{itemize}
The following formulas hold.
\begin{enumerate}
\item If $0 \le m_1,n_1 \le m_0+n_0$, then $\delta=d^*$.
\item If $0 \le n_1 \le m_0+n_0 < m_1$, then $\delta=d^*+m_1-(m_0+n_0)$.
\item If $0 \le m_1 \le m_0+n_0 < n_1$, then  $\delta=d^*+n_1-(m_0+n_0)$.
\item If $m_0+n_0 < m_1 \le n_1 < q$, then $\delta=d^*+m_1+n_1-2(m_0+n_0)$.
\item If $m_0+n_0 < n_1 \le m_1 < q$, then $\delta=d^*+m_1+n_1-2(m_0+n_0)$.
\item If $m_0+n_0<m_1,n_1$ and $m_1=n_1=q$, then $\delta=d^*+q-(m_0+n_0)$.
\end{enumerate}
\end{theorem} 


\begin{theorem}[\cite{Park}, Theorem 3.5]
\label{3.5}
Let $G=mP_\infty + nP_0$ and let $C^\perp$ be the dual of the code obtained
evaluating the vector space $L(G)$ on the points of $B$. Denote by $\delta$ its
minimum distance and write
\begin{eqnarray*}
m&=&m_0(q+1)+m_1, \ \ \  0 \le m_1 \le q, \\
n&=&n_0(q+1)+n_1, \ \ \ \ 0 \le n_1 \le q. 
\end{eqnarray*}
Suppose that $G$ satisfies either
\begin{itemize}
\item[(a)] $\deg(G)< \deg(K)$ or
\item[(b)] $\deg(K) \le \deg(G) \le \deg(K)+q$ and $G \sim sP_\infty$ or $G \sim
tP_0$, for some $s,t \in \numberset{Z}$.
\end{itemize}
Then $\delta=m_0+n_0+2$.
\end{theorem}



The Hermitian curve $X$ has a very particular algebraic geometry. It is deeply
studied, for instance, in \cite{AC}. Let us recall some basic properties (other
important facts will be stated in Section \ref{sec2}). For any $P,Q \in
X(\numberset{F}_{q^2})$ we get a linear equivalence $(q+1)P\sim(q+1)Q$. Moreover, for any
$P \in X(\numberset{F}_{q^2})$ we have an isomorphism of shaves
$\mathcal{O}_X(1)\cong\mathcal{L}((q+1)P)$, the latter one being the invertible sheaf
associated to the divisor $(q+1)P$ on $X$. By these geometric facts we easily
deduce that for any pair of integers $(m,n)$ such that $m+n>0$ there exists a
tern of integers $(d,a,b)$ such that $d>0$, $0 \le a,b \le q$ and
$L(mP_\infty+nP_0) \cong H^0(X,\mathcal{O}_X(d)(-E))$, where $E:=aP_\infty+bP_0$. Hence
$C_{m,n}$ turns out to be isometric to the two-point code obtained evaluating
the vector space $H^0(X,\mathcal{O}_X(d)(-E))$ on $B$, here denoted by $C(d,a,b)$.

\begin{remark} \label{remar2}
  By the 2-transitivity of $\mbox{Aut}(X)$ we may also assume $a \le b$ by
permuting (if necessary) $P_0$ and $P_\infty$. Moreover, note that if $b=0$ then
$C(d,a,b)$ is not a code of interest. Indeed, $C(d,a,0)$ is a one-point code
whose parameters can be easily improved by evaluating $H^0(X,\mathcal{O}_X(d)(-E))$ on
$X(\numberset{F}_{q^2})\setminus \{P_\infty\}$ instead of $B$. Hence, from now on, we will
implicitely assume $a \le b$ and  $b \neq 0$ when writing $C(d,a,b)$.
\end{remark}


Recently A. Couvreur found a way to characterize the dual minimum distance of
some geometric Goppa codes in terms of their projective geometry (see
\cite{c3}). We extended his approach in \cite{br} in order to find out the dual
minimum distance of some $m$-points codes on the Hermitian curve. Here we
combine the results by Park and the projective geometry of the Hermitian curve
in order to obtain a geometric description of the minimum-weight codewords of
any $C(d,a,b)^\perp$ code such that $2<d \le q-1$. To be precise, we will show
that the points in their supports obey some particular geometric laws. The
peculiarity of the paper consists in the application of classical tools of
Algebraic Geometry (for instance, the Cohomology of zero-dimensional schemes) to
Coding Theory problems.



\begin{remark}
If $m \in \numberset{Z}_{>0}$ consider the Hermitian one-point code $C_m$ obtained
evaluating the vector space $L(mP_\infty)$ on $X(\numberset{F}_{q^2}) \setminus \{ P_\infty
\}$. Our approach can be easily applied also to this kind of codes, obtaining
(and extending) the results of \cite{SMP}. On the other hand, the methods
applied in the cited paper, based on the particular structure of certain basis
of $L(mP_\infty)$ (see for instance \cite{DK}), cannot be of help in the
two-point case, even if an explicit basis of $L(mP_\infty+nP_0)$ is well-known
for any $m,n \in \numberset{Z}$ (\cite{DK} again). 
\end{remark}



\section{Preliminary results} \label{sec2}
On first, let us state some technical lemmas about the geometry of the Hermitian
curve and zero-dimensional subschemes of $\numberset{P}^2$.

\begin{lemma}
\label{intrette}
Let $X$ be the Hermitian curve. Every line $L$ of $\numberset{P}^2$ either intersects $X$
in $q+1$ distinct $(\numberset{F}_{q^2})$-rational points, or $L$ is tangent to $X$ at a
point $P$ (with contact order $q+1$). In the latter case $L$ does not intersect
$X$ in any other $\numberset{F}_{q^2}$-rational point different from $P$.
\end{lemma}
\begin{proof}
See \cite{pj}, part (i) of Lemma 7.3.2, at page 247.
\end{proof}

\begin{lemma}
\label{intcurve}
Let $X\subseteq \numberset{P}^2$ be the Hermitian curve. Fix an integer $e\in \{2,\dots
,q+1\}$ and $P\in 
 X(\mathbb {F}_{q^2})$. Let $E\subseteq X$ be the divisor $eP$, seen as a closed
degree $e$ subscheme of $\mathbb {P}^2$. Let $L_{X,P}\subseteq \mathbb {P}^2$ be
the tangent line
 to $X$ at $P$. Let $T\subset \mathbb {P}^2$ be any effective divisor
 (i.e. a plane curve, possibly with multiple components)
 of degree $\le e-1$ and containing $E$. Then $L_{X,P}\subseteq T$, i.e.
$L_{X,P}$ is one of the components of $T$.
\end{lemma}

\begin{proof}
Since $L_{X,P}$ has order of contact $q+1\ge e$ with $X$ at $P$, we have
$E\subset L_{X,P}$.
Since $\deg (E)>\deg (T)$ and $E\subseteq T\cap L_{X,P}$, Bezout theorem implies
$L_{X,P}\subseteq T$.
\end{proof}

\begin{lemma} \label{tgline}
 Let $X \subseteq \numberset{P}^2$ be the Hermitian curve and let $P \in X(\numberset{F}_{q^2})$.
Denote by $L_{X,P}$ the tangent line to $X$ at $P$. Let $C \subseteq \numberset{P}^2$ be
any irreducible smooth curve defined over $\numberset{F}_{q^2}$ such that $P \in
C(\numberset{F}_{q^2})$. Let $L_{C,P}$ be the tangent line to $C$ at $P$. Then $2P
\subseteq C$ if and only if $L_{X,P}=L_{C,P}$. 
\end{lemma}
\begin{proof}
 Use both the definition of tangent line and Lemma \ref{intcurve}, with
$T:=L_{C,P}$ and $E:=2P$.
\end{proof}


\begin{lemma}\label{u00.01}
Fix integers $d>0$, $z\ge 2$ and a zero-dimensional scheme $Z\subset \mathbb
{P}^2$ such that
$\deg (Z) =z$.

\begin{enumerate}
\item[(a)] If $z\le d+1$, then $h^1(\mathbb {P}^2,\mathcal {I}_Z(d))=0$.

\item[(b)]\label{strano1} If $d+2 \le z\le 2d+1$, then $h^1(\mathbb
{P}^2,\mathcal {I}_Z(d))>0$ if and only if there
is a line $T_1$ such that $\deg (T_1\cap Z)\ge d+2$.

\item[(c)]\label{strano2} If $2d+2\le z \le 3d-1$, then $h^1(\mathbb
{P}^2,\mathcal {I}_Z(d))>0$ if and only if
either there
is a line $T_1$ such that $\deg (T_1\cap Z)\ge d+2$, or there is a conic $T_2$
such that $\deg (T_2\cap Z) \ge 2d+2$.

\item[(d)]\label{strano3} Assume $z=3d$. Then $h^1(\mathbb {P}^2,\mathcal
{I}_Z(d))>0$ if and only if
either there
is a line $T_1$ such that $\deg (T_1\cap Z)\ge d+2$, or there is a conic $T_2$
such that $\deg (T_2\cap Z) \ge 2d+2$, or there is a plane cubic
$T_3$ such that $Z$ is the complete intersection of $T_3$ and a plane curve of
degree
$d$.

\item[(e)]\label{ultimo} Assume $z \le 4d-5$.  Then $h^1(\mathbb {P}^2,\mathcal
{I}_Z(d))>0$ if and only if
either there
is a line $T_1$ such that $\deg (T_1\cap Z)\ge d+2$, or there is a conic $T_2$
such that $\deg (T_2\cap Z) \ge 2d+2$, or there are $W\subseteq Z$ with $\deg
(W)=3d$ and plane cubic
$T_3$ such that $W$ is  the complete intersection of $T_3$ and a plane curve of
degree
$d$, or there is a plane cubic $C_3$ such that $\deg (C_3\cap Z)\ge 3d+1$.
\end{enumerate}
\end{lemma}
\begin{proof}
 See \cite{br}, Lemma 2.
\end{proof}

The previous lemma implies the following useful result.
\begin{lemma}\label{u1}
Fix an integer $m>0$ and a zero-dimensional scheme $Z\subset \mathbb
{P}^2$ such that $2 \le \deg(Z) \le \max\{2m+2, 3m,4m-5\}$. If $h^1(\mathbb
{P}^2,\mathcal {I}_Z(m))>0$ there must exist a subscheme $W\subseteq Z$ such
that one of
the following cases occurs.
\begin{enumerate}
\item[(a)] $\deg (W)=m+2$ and $W$ is contained in a line;
\item[(b)] $\deg (W) = 2m+2$ and $W$ is contained in a conic;
\item[(c)] $\deg (W) =3m$ and $W$ is the complete intersection of a
curve
of degree $3$ and a curve of degree $m$;
\item[(d)] $\deg (W)=3m+1$ and $W$ is contained in a cubic curve.
\end{enumerate}
\end{lemma}
\begin{proof}
 Apply the ``~only if~'' part of Lemma \ref{u00.01}.
\end{proof}



\begin{lemma}\label{e1}
Fix a smooth plane curve $X\subset \mathbb {P}^2$, an integer $d>0$, a
zero-dimensional
scheme $E\subset X$ and a finite
subset $B\subset X$ such that $B\cap E_{\mbox{red}}=\emptyset$. Let $C$ be the
code on $X$ obtained
evaluating the vector space $H^0(X, \mathcal {O}_X(d)(-E))$ at the points of
$B$. Set
$c:= \deg (X)$.
Assume $\sharp (B) > dc$. Set $n:= \sharp (B)$ and $k:= h^0(X,\mathcal
{O}_X(d))-\deg (E) $, where
$h^0(X,\mathcal {O}_X(d)) =\binom{d+2}{2}$ if $d < c$ and $h^0(X,\mathcal
{O}_X(d))=\binom{d+2}{2}
-\binom{d-c+2}{2}$ if $d \ge c$. Then $C$ is an $[n,k]$-code and the minimum
distance
of $C^{\bot}$ is the minimal cardinality, say $s$, of a subset $S \subseteq B$
of $B$ such that
$h^1(\mathbb {P}^2,\mathcal {I}_{S\cup E}(d)) >h^1(\mathbb {P}^2,\mathcal
{I}_E(d))$. A given codeword of $C^{\bot}$
has weight $s$ if and only if it is supported by $S\subseteq B$ such that
$\sharp (S) = s$ and $h^1(\mathbb {P}^2,\mathcal {I}_{E\cup S}(d)) >h^1(\mathbb
{P}^2,\mathcal {I}_E(d))$.
\end{lemma}

\begin{proof}
The computation of $h^0(X,\mathcal {O}_X(d))$ is well-known. We impose that $B$
does
not intersect the support of $E$. The case $E=\emptyset$ is a particular case of
\cite{c3}, Proposition
3.1. In the general case notice that $C$ is obtained evaluating
a family of homogeneous degree $d$ polynomials (the ones vanishing on the scheme
$E$)
at the points of $B$.
Since $X$ is projectively normal, the restriction map
\begin{eqnarray*}
\rho _d: H^0(\mathbb {P}^2,\mathcal {O}_{\mathbb {P}^2}(d)) \to H^0(X,\mathcal
{O}_X(d))
\end{eqnarray*} is surjective.
As a consequence the restriction map 
\begin{eqnarray*}
\rho _{d,E}: H^0(\mathbb {P}^2,\mathcal {I}_E(d)) \to H^0(X,\mathcal
{O}_X(d)(-E))
\end{eqnarray*} is surjective. Hence a finite subset $S\subseteq X\setminus
E_{\mbox{red}}$
imposes independent condition to the vector space $H^0(X,\mathcal {O}_X(d)(-E))$
if and only
if $S$ imposes independent conditions to $H^0(\mathbb {P}^2,\mathcal {I}_E(d))$.
On the other hand, $S$ imposes independent conditions to $H^0(\mathbb
{P}^2,\mathcal {I}_E(d))$
if and only if $h^1(\mathbb {P}^2,\mathcal {I}_{E\cup S}(d)) =h^1(\mathbb
{P}^2,\mathcal {I}_E(d))$
(here we use again that $S\cap E=\emptyset$).
\end{proof}
\begin{remark} \label{remar}
 In the notations of Lemma \ref{e1} a given minimum-weight codeword of
$C^\perp$, whose support is $S$, trivially satisfies the condition
$h^1(\numberset{P}^2,\mathcal{I}_{E \cup S}(d))>0$, which is described by Lemma \ref{u1}.
\end{remark}



\begin{proposition} \label{pr1}
 Fix
integers $d>2$, $a_i \in \{0,...,q\}$ for $i=1,2$ with $a_1 \le a_2$. Let $r$ be
(if it exists) the maximum integer $i\le 2$
such
that $a_i\le d-2+i$. Otherwise set $r:=0$.
Set $d':= d-2+r>0$. If $r>0$ set $a'_i:= a_i$ for $i\le r$ and $a'_i:=0$
otherwise. If $r=0$ set $a'_i:=0$ for any $i \in \{1,2\}$. Let $X\subset
\mathbb
{P}^2$ be the Hermitian curve and let $P_1:=P_\infty$, $P_2:=P_0$.  Define $E:=
\sum_{i=1}^2 a_i P_i$. If $r>0$ define $E':= \sum _{i=1}^{2}a'_iP_i$. Otherwise
set $E':=0$. Then the codes obtained evaluating the two vector spaces
$H^0(X,\mathcal{O}_X(d)(-E))$ and $H^0(X,\mathcal{O}_X(d)(-E'))$ on $B=X(\numberset{F}_{q^2})\setminus
\{P_0,P_\infty \}$ are the same code.
\end{proposition}

\begin{proof}
 If $r=2$ then $E=E'$, $d=d'$ and so we have nothing to prove. If $r=1$ then
$d'=d-1$, $a_1'=a_1$, $a_2'=0$ and $E':=a_1P_1$. 
 Consider $E$ and $E'$ as closed subschemes of $\numberset{P}^2$ of degree $a_1+a_2$ and
$a_1$, respectively. Since the restriction map 
\begin{equation*}
H^0(\mathbb {P}^2,\mathcal {O}_{\mathbb {P}^2}(d))
\to H^0(X,\mathcal {O}_X(d))
\end{equation*}
is surjective then the restriction maps
\begin{equation*}
\rho _E: H^0(\mathbb {P}^2,\mathcal {I}_{E}(d)) \to
H^0(X,\mathcal
{O}_X(d)(-E))
\end{equation*} and
\begin{equation*}
\rho _{E'}:H^0(\mathbb {P}^2,\mathcal {I}_{E'}(d))
\to
H^0(X,\mathcal{O}_X(d)(-E'))
\end{equation*} are surjective themselves.
Every tangent line $T_PX$ to $X$ at a point $P\in X(\mathbb {F}_q)$
has order of contact $q+1$ with $X$ at $P$ (Lemma \ref{intrette}) and hence by
Bezout's
theorem
it intersects $X$ only
at $P$. As a consequence $T_{P_2}X\cap E$ has degree $a_2$. Since $a_2>d$ we get
that
every degree $d$
homogeneous form vanishing on $E$ vanishes also on the line $T_{P_2}X$,
i.e. it
is divided by the equation of $T_{P_2}X$. Since $\rho _E$ and
$\rho _{E'}$ are surjective and $B\cap T_{P_2}X =\emptyset$,
we get that
the codes obtained 
evaluating $H^0(X,\mathcal {O}_X(d)(-E))$ and $H^0(X,\mathcal
{O}_{X}(d')(-E'))$, respectively, are in fact the same code. In case $r=0$
repeat the same procedure of case $r=1$ with $a_2$ and $d$, and then with $a_1$
and $d-1$.
\end{proof}

Combining Lemma \ref{u1}, Remark \ref{remar} and Proposition \ref{pr1} we get
the following result.
\begin{proposition}\label{pr2}
Fix integers $d>2$, $a_i \in \{0,...,q\}$ for $i=1,2$. Let $r$, $d'$, $E$ and
$E'$ be as in Proposition \ref{pr1}. Denote by $C:=C(d,a_1,a_2)$ the code
obtained evaluating the vector space $H^0(X,\mathcal{O}_X(d)(-E))$  on
$B=X(\numberset{F}_{q^2})\setminus \{P_0,P_\infty \}$ and let $C^\perp$ be its dual code.
Let $S \subseteq B$ be the support of a minimum-weight codeword of $C^\perp$ and
assume $\deg(E')+\sharp(S) \le \max\{2d'+2,3d',4d'-5\}$. There exists a
zero-dimensional scheme $W \subseteq E' \cup S$ which satisfies one of the
following properties.
\begin{enumerate}
\item[(a)] $\deg (W)=d'+2$ and $W$ is contained in a line;
\item[(b)] $\deg (W) = 2d'+2$ and $W$ is contained in a conic;
\item[(c)] $\deg (W) =3d'$ and $W$ is the complete intersection of a
curve
of degree $3$ and a curve of degree $d'$;
\item[(d)] $\deg (W)=3d'+1$ and $W$ is contained in a cubic curve.
\end{enumerate}
\end{proposition}

 
\section{Geometry of minimum-weight codewords}

Following Park's Theorems \ref{3.3} and \ref{3.5}, consider these three groups
(say (G1), (G2) and (G3)) of two-point Hermitian codes $C(d,a,b)$ (assume always
 $0 \le a \le b \le q$ and $b \neq 0$ as in Remark \ref{remar2}). We will denote
by $K$ any canonical ($\numberset{F}_{q^2}$-rational) divisor on $X$.
\begin{enumerate}
\item[(G1)] The codes $C(d,a,b)$ such that $0<d(q-1)-a-b<\deg(K)$.
\label{gruppi}
\item[(G2)] The codes $C(d,a,b)$ such that $\deg(K) \le d(q-1)-a-b \le
\deg(K)+q$ and $d(q+1)P_\infty-aP_\infty-bP_0$ is not linearly equivalent to a
multiple of $P_\infty$ nor to a multiple of $P_0$.
\item[(G3)] The codes $C(d,a,b)$ such that $\deg(K) \le d(q-1)-a-b \le
\deg(K)+q$ and $d(q+1)P_\infty-aP_\infty-bP_0$ is linearly equivalent to a
multiple of $P_\infty$ or to a multiple of $P_0$ .
\end{enumerate}
Here we apply the results of the previous section in order to describe the
minimum-weight codewords of the dual one of a Hermitian two-point code. 

\begin{theorem} \label{1gruppo}
Let $C:=C(d,a,b)$ be a code of group (G1), with $d>2$, $0 \le a \le b \le q$ and
$b \neq 0$. Set $a_1:=a$, $a_2:=b$  and define $r$, $d'$, $E$, $E'$ as in
Proposition \ref{pr1}.
If $a \ne 0$ then the minimum distance of $C^\perp$ is $d$. Let
$\{P_1,...,P_d\}$ be the support of a minimum-weight codeword.
\begin{enumerate}
 \item[(a)] If $r=0$ then $P_1,...,P_d$ are collinear.
\item[(b)] If $r=1$ then $P_\infty,P_1,...,P_d$ are collinear.
\item[(c)] If $r=2$ and $d>3$ then $P_1,...,P_d$ are collinear.
\item[(d)] If $r=2$ and $d=3$ then
\begin{enumerate}
\item[(d.1)] either $P_1,...,P_d$ are collinear,
\item[(d.2)] or they lie on the union of the line through $P_0$ and $P_\infty$
and a smooth conic which is tangent to $X$ at both $P_0$ and $P_\infty$,
\item[(d.3)] or they lie on an irreducible cubic which is tangent to $X$ at both
$P_0$ and $P_\infty$.
\end{enumerate}
\end{enumerate}
If $a=0$ then the minimum distance of $C^\perp$ is $d+1$ and $r \neq 0$. Let
$\{P_1,...,P_{d+1}\}$ be the support of a minimum-weight codeword of $C^\perp$.
\begin{enumerate}
\item[(a)] If $r=1$ then $P_1,...,P_d$ are collinear.
\item[(b)] If $r=2$ then $P_0,P_1,...,P_d$ are collinear.
\end{enumerate}
\end{theorem}


\begin{proof}
Denote by $L_{X,P_0}$, $L_{X,P_\infty}$ the tangent lines to $X$ at $P_0$ and
$P_\infty$ (respectively). Denote by $L_{0,\infty}$ the line through $P_0$ and
$P_\infty$. Let us consider first the case $a \ne 0$. Keep on mind the linear
equivalence
\begin{equation*}
 d(q+1)P_\infty-aP_\infty-bP_0 \sim ((d-2)(q+1)+(q+1-a))P_\infty+(q+1-b)P_0
\end{equation*}
and apply Theorem \ref{3.5} with $m_0:=d-2$, $m_1:=q+1-a$, $n_0:=0$ and
$n_1:=q+1-b$, obtaining that the minimum distance of $C^\perp$ is $m_0+n_0+2=d$
(here we used $b \neq 0$). Let $S:=\{ P_1,...,P_d\} \subseteq B$ be the support
of a minimum-weight codeword of $C^\perp$. If $r=0$ then $d'=d-2$. Since
$\deg(E')+\sharp(S)=d'+2$, Proposition \ref{pr2} applies: there exists a
subscheme $W \subseteq \left\{P_1,...,P_d \right\}$ of degree $d'+2=d-2+2=d$
contained in a line, and we are done. If $r=1$ then $a \leq d-1 <b$ and
$d':=d-2+1=d-1$. Since $\deg(E')+\sharp(S) \le (d-1)+d=2d-1=2(d'+1)-1=2d'+1$,
Proposition \ref{pr2}  applies and there exists a subscheme $W \subseteq
aP_\infty \cup \{P_1,...,P_d \}$ of degree $d+1$ and contained in a line $L$. It
is immediately seen that $P_\infty \in W$ and $L \neq L_{X,P_\infty}$  (since $a
\le d-1$ and Lemma \ref{intrette} applies). Hence the points $P_\infty,
P_1,...,P_d$ are collinear.
If $r=2$ then $d'=d$ and $a \leq b \leq d$. In this case $E'=E=aP_\infty+bP_0$.
We have $\deg(E)+\sharp(S)\le 3d$ and then Proposition \ref{pr2} applies: either
there exists a subscheme $W \subseteq aP_\infty+bP_0 \cup \{ P_1,...,P_d \}$ of
degree $d+2$ and contained in a line $L$, or there exists a subscheme $W
\subseteq aP_\infty+bP_0 \cup \{ P_1,...,P_d \}$ of degree $2d+2$ and contained
in a conic $T$, or there exists a subscheme  $W \subseteq aP_\infty+bP_0 \cup \{
P_1,...,P_d \}$ of degree $3d$  which is the complete intersection of a curve of
degree 3 and one of degree $d$.
In the former case, since $\deg (W)=d+2$, we see that both $P_\infty$ and $P_0$
must lie on $L$, unless $L=L_{X,P_\infty}$ or $L=L_{X,P_0}$ , which is forbidded
by Lemma \ref{intrette}. All the points $P_1,...,P_d$ lie on the line
$L_{0,\infty}$.
Now assume $\deg(W)=2d+2$ and $W$ contained in a conic $T$. The sum of the
multiplicities of $P_\infty$ and $P_0$ in $W$, say $e_W(P_\infty)$ and
$e_W(P_0)$, must satisfy $e_W(P_\infty)+e_W(P_0) \ge (2d+2)-d=d+2>4$. Since
$e_W(P_\infty) \le d$ and $e_W(P_0) \le d$ we deduce (Lemma \ref{intcurve})
$T=L_{X,P_0} \cup L{X,P_0}$, which contradicts Lemma \ref{intrette}. 
Finally, assume $\deg(W)=3d$. Then $a=b=d>2$ and $dP_\infty+dP_0+P_1,...,P_d$ is
the complete intersection of a cubic curve $C$ and a curve of degree $d$. If
$d>3$ then Lemma \ref{intcurve} implies $L_{X,P_\infty} \subseteq C$ and
$L_{X,P_0} \subseteq C$. By Lemma \ref{intrette} $P_1,...,P_d$ are collinear
again (they lie on the line $C-L_{X,P_\infty} -L_{X,P_0}$). If $d=3$ then it
occurs exactly one of the following cases.
\begin{itemize}
 \item The cubic $C$ is reducible and a degree one component of $C$, say $L$, is
tangent to $X$.
\begin{itemize}
\item If $L \neq L_{X,P_\infty}$ and $L \neq L_{X,P_0}$ then Lemma
\ref{intrette} and Lemma \ref{intcurve} imply (keeping on mind our assumption
$d>2$) $L_{X,P_0}, L_{X,P_\infty} \subseteq C - L$. This is forbidden by Lemma
\ref{intrette}. 

\item  If $L=L_{X,P_\infty}$ (or $L=L_{X,P_0}$) then ($d>2$) Lemma
\ref{intcurve} and Lemma \ref{intrette} imply also $L_{X,P_0} \subseteq C-
L_{X,P_\infty}$ (or $L_{X,P_\infty} \subseteq C- L_{X,P_0}$). Hence
$P_1,...,P_d$ lie on the line $C -L_{X,P_\infty} - L_{X,P_0}$. 
\end{itemize}
\item The cubic $C$ is reducible and no degree one component of it is tangent to
$X$. Let $L\subseteq C$ be a line. By hypothesis it is not tangent to $X$.
\begin{itemize}
\item If $L$ is not the line through $P_0$ and $P_\infty$ then $3P_\infty
\subseteq C- L$ or $3P_0 \subseteq C - L$ (or both). In any case, since
$2P_\infty+2P_0 \subseteq C - L$, we have $L_{X,P_\infty} \subseteq C$ and
$L_{X,P_0} \subseteq C$ (here we used Lemma \ref{intrette} again). As a
consequence, $P_1,...,P_d$ lie on $L$ (Lemma \ref{intrette}).
\item If $L$ is the line through $P_0$ and $P_\infty$ then either $C=L\cup
L_{X,P_0} \cup L_{X,P_\infty}$ and $P_1,...,P_d$ lie on $L$ (by Lemma
\ref{intrette}), or $C=L \cup T$, and $T$ is a smooth conic tangent to $X$ at
both $P_0$ and $P_\infty$ (remember that $2P_\infty+2P_0 \subseteq C - L$ and
Lemma \ref{tgline} holds).
\end{itemize}
\item The cubic $C$ is an irreducible curve and tangent (by Lemma \ref{tgline})
to $X$ at both $P_0$ and $P_\infty$.
\end{itemize}
Now consider the simpler case $a=0$. Theorem \ref{3.5} proves that the minimum
distance is $d+1$.
If $r=0$ then $b>d$ and $0=a>d-1$, but we assumed $d>2$.
If $r=1$ then  $\deg(E') +\sharp(S) =\sharp(S)=d+1=d'+2$ and Proposition
\ref{pr2} applies: there exists subscheme $W \subseteq \{ P_1,...,P_{d+1}\}$ of
degree $d'+2=d+1$ and contained in a line. In particular $P_1,...,P_{d+1}$ turns
out to be collinear.
If $r=2$ then $b \le d$, $E'=E$ and $d'=d$. We see that $\deg(E)+\sharp(S) \le
d+(d+1)=2d+1$ and Proposition \ref{pr2} applies: there exists a subscheme $W
\subseteq bP_0 \cup \{P_1,...,P_d \}$ of degree $d+2$ and contained in a line
$L$. By Lemma \ref{intrette}, $L$ cannot be the tangent line to $X$ at $P_0$. As
a consequence, $P_0$ appears is $L$ with multiplicity exactly one and so
$P_0,P_1,...,P_{d+1}$ are collinear. 
\end{proof}

To complete our analysis we need the following geometric result.
\begin{lemma} \label{gonalit}
 Let $X$ be the Hermitian curve. Let $P,Q \in X(\numberset{F}_{q^2})$ with $P \neq Q$.
Assume $nP \sim nQ$ for a certain $0 \le n \le q$. Then $n=0$.
\end{lemma}
\begin{proof}
 Assume $n \neq 0$. Let $h:=\mbox{gcd}(n,q+1) \le n \le q$. Write $h=\alpha
n+\beta(q+1)$. We deduce the linear equivalences $hP \sim \alpha n
P+\beta(q+1)P$ and $hQ \sim \alpha n Q+\beta(q+1)Q$. Since $nP \sim nQ$ then
$\alpha n P \sim \alpha n Q$. Hence $hP -\beta(q+1)P \sim hQ -\beta(q+1)Q$.
Since $\beta(q+1)P \sim \beta(q+1)Q$ (as described in Section \ref{intr}) we
have $hP \sim hQ$, with $1 \le h \le q$ such that $h$ divides $q+1$. This
implies that the gonality of $X$ is at most $h<q$, but it is well-known that $X$
has gonality exactly $q$ (\cite{AC}, Remark 9.50 (iii) at page 357). This
implies $n=0$.
\end{proof}



\begin{remark} \label{no3}
 If $C(d,a,b)$ is in group (G3) then we have either $d(q+1)-aP_\infty-bP_0 \sim
s P_\infty$ for a certain $s \in \numberset{Z}$,
or $d(q+1)-aP_\infty-bP_0 \sim t P_0$ for a certain $t \in \numberset{Z}$. In the former
case $s=d(q+1)-a-b$ and hence $bP_0 \sim bP_\infty$. Since $b \le q$ we deduce
(Lemma \ref{gonalit}) $b=0$, which contradicts our assumptions (Remark
\ref{remar2}). In the latter case we get $d(q+1)P_\infty-aP_\infty-bP_0 \sim
d(q+1)P_0-aP_0-bP_0$.
Since $(q+1)P_\infty \sim (q+1)P_0$ (see Section \ref{intr}) we deduce
$aP_\infty \sim aP_0$ and then (Lemma \ref{gonalit} again) $a=0$.
\end{remark}



Since we assumed $b \neq 0$ (Remark \ref{remar2}) and a canonical divisor $K$ on
$X$ has degree\footnote{Any smooth plane curve of degree $c$ has genus
$g(X)=(c-1)(c-2)/2$. The degree of a canonical divisor $K$ on $X$ is always
$2g(X)-2$. Remember that the Hermitian curve has degree $q+1$.} $q^2-q-2$, we
immediately see that if $d<q-1$ then $C(d,a,b)$ is of group (G1), for any choice
of $0 \le a \le b \le q$. If $d=q-1$ and $a \neq 0$ then $C(d,a,b)$ could be
either in group (G1), or in group (G2). Indeed, $(q-1)(q+1)-a-b \le q^2-2$ for
any $1 \le a \le b \le q$ and group (G3) is excluded by Remark \ref{no3}.
On the other hand, $C(q-1,0,b)$ is always in group (G3), because $q^2-q-2 \le
(q-1)(q+1)-b \le q^2-2$ for any $1 \le b \le q$. This proves that in order to
complete our study of case $d=q-1$ we can analyse separately the codes
$C(q-1,a,b)$ of group (G2), which satisfy $1 \le a \le b \le q$, and the codes
of the form $C(q-1,0,b)$ (obviously $b \le q$), which are in group (G3). Note
that in case $d=q-1$ our usual assumption $d>2$ is equivalent to $q > 3$. 



\subsection{Codes $C(q-1,a,b)$ of group (G2)}
In the following Remark \ref{car2} we give a characterization of the codes
$C(q-1,a,b)$ of group (G2).
\begin{remark}\label{car2}
 A code $C(q-1,a,b)$ of group (G2) satisfies (combining Remark \ref{remar2} and
Remark \ref{no3}) $1 \le a \le b \le q$. The condition of being in group (G2) is
\begin{equation*}
 q^2-q-2 \le (q-1)(q+1)-a-b \le q^2-2,
\end{equation*}
which is equivalent to $a+b \le q+1$ (the right-hand side condition always
holds). 
\end{remark}
Now consider the linear equivalence
\begin{eqnarray*}
(q-1)(q+1)P_\infty-aP_\infty-bP_0 &\sim& (q-2)(q+1)P_\infty
+(q+1-a)P_\infty-bP_0 \\
&\sim& K + (q+1-a)P_\infty-bP_0
\end{eqnarray*}
and apply Theorem \ref{3.3} with $m=q+1-a$, $n=-b$, $m_0=1$, $m_1=a$, $n_0=0$
and $n_1=b$. Since we assumed $a \le b$ and Remark \ref{car2} holds, only Park's
cases (1), (3) and (4) of Theorem \ref{3.3} are of our interest.  The dual
minimum distance, say $\delta(q-1,a,b)$, of a code $C(q-1,a,b)$ of group (G2)
can be read in Table \ref{md2}.

\begin{table}[h!]
\centering
\small{
\begin{center}
\begin{tabular}[h!]{|p{2.6cm}|p{2.4cm}|p{2.4cm}|p{2.4cm}|}
\hline
Park's case $\rightarrow$ & (1) & (3) & (4)  \\
\hline
\hline
$\delta(q-1,a,b)$ & $q+1-a-b$ & $q-a$ & $q-1$  \\
\hline
\end{tabular}
\end{center}
}
\caption{Dual minimum distance of a code $C(q-1,a,b)$ of group (G2).}
\label{md2}
\end{table}



\begin{theorem} \label{te2}
 Let $C(q-1,a,b)$ be a code of group (G2) (conditions $d=q-1>2$, $1 \le a \le b
\le q$ and $a+b \le q+2$ will be implicitely assumed). Let
$\delta:=\delta(q-1,a,b)$ be the dual minimum distance of $C(q-1,a,b)$ and let
$S:=\{ P_1,...,P_\delta\}$ be the support of a minimum-weight codeword. There
occurs one of the following situations.
\begin{enumerate}
 \item[(a)] $P_1,...,P_\delta$ lie on the line through $P_0$ and $P_\infty$.
\item[(b)] $P_\infty, P_1,...,P_\delta$ are collinear.
\item[(c)] $P_0, P_1,...,P_\delta$ are collinear.
\end{enumerate}
\end{theorem}
\begin{proof}
Let us examine separately cases (1), (3) and (4) of Theorem \ref{3.5}. In the
notations of Proposition \ref{pr1} set $a_1:=a$, $a_2:=b$, $d:=q-1$,
$E:=aP_\infty+bP_0$ and define $E'$ and $d'$ as in the statement.
\begin{enumerate}
 \item If $C(q-1,a,b)$ is described by case (1) of Theorem \ref{3.5} then $a,b
\le 1$. Hence $a=b=1$ and  the  dual minimum distance is $\delta=q-1$ (see Table
\ref{md2}). Since $d>2$ we have $E'=E$ and $d'=d$. Note that
$\deg(E')+\sharp(S)=q-1+2=q+1=d+2$ and Proposition \ref{pr2} applies: there
exists a subscheme $W \subseteq P_\infty+P_0 \cup \{ P_1,...,P_{q-1}\}$ of
degree $q+1$ and contained in a line. This means that $P_1,...,P_{q-1}$ lie on
the line through $P_\infty$ and $P_0$.
\item If $C(q-1,a,b)$ is described by case (3) of Theorem \ref{3.5} then $0 \le
a \le 1 < b$. Hence $a=1$. 
\begin{enumerate}
\item[(2.i)] If $b<q$ then $d'=d$, $E'=E$ and the dual minimum distance is
$\delta=q-a=q-1$. Note that $\deg(E')+\sharp(S)=(q-1)+(1+b)=q+b=d+1+b$ and
Proposition \ref{pr2} applies: there exists a subscheme $W \subseteq
P_\infty+bP_0 \cup \{ P_1,...,P_{q-1}\}$ of degree $q+1$ and contained in a
line\footnote{The case $\deg(W)\ge 2d+2=2q$ implies $b \ge q$, which contradicts
our hypothesis on $b$.}. By Lemma \ref{intrette}, $P_0$ must appear in $W$ with
multiplicitly exactly one and $P_1,...,P_{q-1}$ lie on the line through
$P_\infty$ and $P_0$.
\item[(2.ii)] If $b=q=d+1$ then $d'=d-1$, $E'=P_\infty$ and the dual minimum
distance is $\delta=q-a=q-1$. Since $\deg(E')+\sharp(S)=q-1+1=q=d'+2$,
Proposition \ref{pr2} applies and there exists a subscheme $W \subseteq P_\infty
\cup \{P_1,...,P_{q-1}\}$ of degree $d'+2=q$ and contained in a line. In other
words $P_\infty,P_1,...,P_d$ turns out to be collinear.
\end{enumerate}
\item If $C(q-1,a,b)$ is described by case (4) of Theorem \ref{3.5} then $1 < a
\le b <q$. In particular $b \le d$, $d'=d$ and $E'=E$. The dual minimum distance
is $\delta=q-1$ and $\deg(E)+\sharp(S)=q-1+a+b=d+a+b \le d+(q+1)=2d+2$. Hence
Proposition \ref{pr2} applies: either there exists a subscheme $W \subseteq
aP_\infty+bP_0 \cup \{P_1,...,P_{q-1} \}$ of degree $q+1$ and contained in a
line $L$, or there exists a subscheme $W \subseteq aP_\infty+bP_0 \cup
\{P_1,...,P_{q-1}\}$ of degree $2q$ and contained in a conic $T$. In the former
case Lemma \ref{intrette} implies that $P_\infty$ and $P_0$ appear in $W$ with
multiplicity exactly one and then $P_1,...,P_{q-1}$ lie on the line through
$P_\infty$ and $P_0$. In the latter case it's immediately seen that
$W=aP_\infty+bP_0+\sum_{i=1}^{q-1}P_i$. Since $\deg(W)=2q$ it must be $a+b=q+1$.
Since our general assumption $d=q-1>2$ holds, it follows $\min \{a,b \}>2$ and
then $L_{X,P_\infty} \subseteq T$ or $L_{X,P_0} \subseteq X$ (Lemma
\ref{intcurve}). In any case (Lemma \ref{intrette}) either $P_0,P_1,...,P_{q-1}$
are collinear, or $P_\infty,P_1,...,P_{q-1}$ are collinear.
\end{enumerate}
This concludes our proof.
\end{proof}

\subsection{Codes $C(q-1,0,b)$ of group (G3)}
If $C(q-1,0,b)$ is a code of group (G3) (i.e. $d=q-1>2$,  $a=0$, $1 \le b \le
q$) then its minimum distance, $\delta(q-1,0,b)$, is given by Theorem \ref{3.5}.
Indeed, choose $m=(q-1)(q+1)$, $m_0=q-1$, $m_1=0$, $n=-b$, $n_0=-1$, $n_1=q+1-b$
and compute $\delta(q-1,0,b)=q$. Let $S:= \{P_1,...,P_q \}$ be the support of a
minimum-weight codeword. In the notations of Proposition \ref{pr1} set $a_1:=0$,
$a_2:=b$, $E:=bP_0$ and define $d'$ and $E'$ as in the statement. If $b \le
d=q-1$ then $E'=E$ and $\deg(E)+\sharp(S)\le q-1+q=2q-1=2d+1$. Hence Proposition
\ref{pr2} applies and gives a zero-dimensional scheme $W \subseteq bP_0 \cup \{
P_1,...,P_q\}$ of degree $q+1$ and contained in a line. By Lemma \ref{intrette}
the multiplicity of $P_0$ in $W$ must be exactly one and then $P_0,P_1,...,P_q$
lie on $L$.
If $d=q$ then $E'=0$ and $d'=d-1=q-2$. Since $\deg(E')+\sharp(S)=q=d+1$
Proposition \ref{pr2} applies: there exists a $W \subseteq \{ P_1,...,P_q\}$ of
degree $d'+2=(d-1)+2=q$ and contained in a line. It follows that $P_1,...,P_q$
are collinear.

Let us summarize the previous analysis in the following concise result.
\begin{theorem} \label{te3}
 Consider a code $C(q-1,0,b)$ of group (G3). Its dual minimum distance is $q$.
Let $\{ P_1,...,P_q\}$ be the support of a minimum-weight codeword. Then
$P_1,...,P_q$ are collinear. Moreover, if $b \le q-1$ then even
$P_0,P_1,...,P_q$ are collinear.
\end{theorem}

\section{Computational Examples}
If $q$ is quite small our results can be checked by writing simple
\texttt{MAGMA} programs (\url{http://magma.maths.usyd.edu.au}). Here you are
some outputs.
\begin{example} \label{exxx}
 Let $q=4$ ($q^2=16$) and let $X$ be the Hermitian curve defined over $\numberset{F}_{16}$
by the affine equation $y^4+y=x^5$. Consider the Hermitian two-point code $C$
obtained evaluating the vector space $L(10P_\infty+3P_0)$ on
$X(\numberset{F}_{16})\setminus \{P_\infty,P_0\}$ (here $P_\infty$ and $P_0$ are those
defined in Section \ref{intr}). Write $10P_\infty+3P_0 \sim
d(q+1)P_\infty-aP_\infty-bP_0$ with $(d,a,b)=(3,0,2)$. Note that $0 \le a \le b
\le q$ and $d=q-1$. Hence our code $C$ is in fact the code $C(3,0,1)$, with
$q=4$, described by Theorem \ref{te3}. If $\numberset{F}_{16}= \langle \alpha \rangle$ and
$\alpha^4+\alpha+1=0$ then a (random) minimum-weight codeword of $C^\perp$ is
\begin{eqnarray*}
 &(& 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
0, 0, 0, 0, 1, 0, 0, \\ & & 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, \alpha^3, 0,
0, 0, 0, 0, 0, 0, 0, 0, \alpha^{12}, 0, 0, 0, \alpha^9, 0, 0, 0 \ \ )
\end{eqnarray*}
with support made of the points of affine coordinates
\begin{equation*}
 P_1:=(\alpha^7, \alpha^7),\ \ P_2:=(\alpha^{11}, \alpha^{11}),\ \
P_3:=(\alpha^{13}, \alpha^{13}),\ \ P_4:=(\alpha^{14}, \alpha^{14}).
\end{equation*}
Note that the points are in fact four (Table \ref{md2} says that the dual
minimum distance of $C$ has to be 4) and they lie (together with $P_0$) on the
line defined over $\numberset{F}_{16}$ by the homogeneous equation $x-y=0$, as described in
Theorem \ref{te3}.
\end{example}

\begin{example}
 In the same notations of Example \ref{exxx} choose $q=5$ and $(d,a,b)=(4,1,1)$.
Note that $d=q-1$ and $b=1$. Hence $C(4,1,1)$ (with $q=5$) is described by
Theorem \ref{te2}. From the proof we get that the dual minimum distance must be
$q-1=4$ and points $P_1$, $P_2$, $P_3$, $P_4$ of the support of a minimum-weight
codeword must lie on the line through $P_0$ and $P_\infty$ (whose equation is
$x=0$). If $\numberset{F}_{125}=\langle \beta \rangle$ (with $\beta^2+4\beta+2=0$) and we
compute the support of a (random) minimum-weight codeword of $C(4,1,1)^\perp$ we
find out the points of affine coordinates
\begin{equation*}
 P_1:=(0,\beta^3), \ \ P_2:=(0,\beta^9), \ \ P_3:=(0,\beta^{15}), \ \
P_4:=(0,\beta^{21}).
\end{equation*}
As one can easily see, they all lie on the line of equation $x=0$. 
\end{example}



