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\title[Codes from surfaces with small Picard number]
{Codes from surfaces with small Picard number}

\author
[John B. Little/joint work with Hal Schenck]
{John B. Little/joint work with Hal Schenck\\
--\\
College of the Holy Cross/University of Illinois\\
--\\
Fq12 -- Saratoga Springs}

\date[July 17, 2015]
{July 17, 2015}

\begin{document}

\begin{frame}
  \titlepage
\end{frame}

%\begin{frame}
%{\bf Outline}
%
%\tableofcontents
%
%\end{frame}



\section{Evaluation codes from algebraic varieties}

\begin{frame}
{\bf Evaluation codes}

\begin{itemize}
\item $X$ an algebraic variety over $\F_q$,
$\mathcal{S} = \{P_1,\ldots,P_n\} \subseteq X(\F_q)$,
$\mathcal{L}$ a vector space of functions on $X$ with all $f(P_i)$ defined.
\item The image of the \emph{evaluation map} 
\begin{eqnarray*}
ev : \mathcal{L} &\to& \F^{\, n}_q\\
       f &\mapsto& (f(P_1),\ldots,f(P_n))
\end{eqnarray*}
is a linear code; $k \le \dim \mathcal{L}$;
$d$ depends on $X$, $\mathcal{S}$, $\mathcal{L}$.
\item \emph{Reed-Solomon codes} $RS(k,q)$ are examples with
$X = \P^1$, $\mathcal{S} = \F_q^{*} \subset \P^1$, and 
$\mathcal{L} = {\rm Span}\{1,x,\ldots,x^{k-1}\} = L((k-1)P_\infty)$ ($k < q$)
(meet the Singleton bound).  
\item AG Goppa codes: $\P^1 \mapsto$ other curves over $\F_q$.
\end{itemize}
\end{frame}

\begin{frame}
{\bf Goal for this work}

\begin{itemize}
\item<1-> What about higher-dimensional varieties $X$?
\item<2-> Some examples have been studied--e.g. {\it projective Reed-Muller codes} from $X = \P^n$
\item<3-> Codes from quadrics, Hermitian varieties, Grassmannians, flag varieties, Deligne-Lusztig varieties
\item<4-> But, still really not much known!
\item<5-> We'll concentrate on $X$ a projective surface and 
{\it Reed-Muller-type} codes with ${\cal S} = X(\F_q)$, ${\cal L} = $
vector space of homogeneous forms of some fixed degree $s$.
\end{itemize}
\end{frame}

\section{The role of small Picard number}

\begin{frame}
{\bf Key issue with these codes; motivating example}

\begin{itemize}
\item One recurrent pattern:  Low weight codewords tend to come from $f$ where
$X \cap {\bf V}(f)$ is \emph{reducible} (possibly if $q >> 0$).
\item Example:  $X$ a quadric surface in $\P^3$.
\item  If $X$ is \emph{hyperbolic}, $|X(\F_q)| = q^2 + 2q + 1$. 
Tangent planes intersect $X$ in reducible curves with $2q + 1$ $\F_q$-points. 
\item But if $X$ is \emph{elliptic}, rulings not defined over $\F_q$ so $|X(\F_q)| = q^2 + 1$, 
and planes meet $X$ in curves with at most $q + 1$ $\F_q$-rational points. 
\item  $s = 1$ codes with $S = X(\F_q)$ have parameters: 
$[q^2+2q+1, 4, q^2]$ (hyperbolic) and $[q^2+1, 4, q^2 - q]$ (elliptic -- better -- equals best known for $q = 8,9$).
\end{itemize}
\end{frame}

\begin{frame}
{\bf \emph{Ansatz} from 2007 thesis of M. Zarzar (UT Austin)}

\begin{itemize}
\item $NS(X)= $ group of {\it $\F_q$-rational 
divisor classes modulo algebraic equivalence} (a finitely-generated abelian group)
\item Example: $X$ an elliptic quadric: $NS(X) = \Z \cdot [H]$, $H$ = a smooth conic plane section;
$X$ a hyperbolic quadric: $NS(X) = \Z [L_1] \oplus \Z [L_2]$ (lines from the two rulings).
\item 
\begin{thm}[Zarzar] If $\deg X = d$ with $(d,{\rm char}(\F_q)) = 1$,
${\rm rank}(NS(X)) = 1$, and $Y$ irreducible over $\F_q$ with $\deg Y < d$,
then $X \cap Y$ is irreducible. \end{thm} \vskip-10pt
\item So ({\bf key idea}) -- good codes (might) come from surfaces $X$ with Picard number $= {\rm rank}\ NS(X) = 1$
(or small).
\end{itemize}
\end{frame}

\section{First examples and role of sectional genus}

\begin{frame}
{\bf A test case -- cubic surface codes}

\begin{itemize}
\item ``Fact 1:''  Over an algebraically closed field, a smooth cubic surface contains
exactly 27 lines, always in a particular highly symmetric configuration.
\item Symmetry group of the 27 lines is a group of order 51840 ($ = W(E_6)$)
\item Frobenius acts as a permutation of the lines
\item There is a complete classification of the conjugacy classes in $W(E_6)$;
the class where Frobenius lies determines the $\F_q$-structure!
\item The 25 possibilities summarized in a 1967 paper of Swinnerton-Dyer (and
in a related table in Manin's book \emph{Cubic Forms}).
\end{itemize}
\end{frame}

\begin{frame}
{\bf An extract from the Swinnerton-Dyer table}

\emph{Exactly five types} of cubics with Picard number $= 1$ ($\Rightarrow$ \emph{no} $\F_q$-rational lines
or conics)

\[
\begin{array}{cccc}
      {\rm Class} & {\rm Perm\ Type\ of\ Frob} & |X(\F_q)|\\ \hline
          C_{10} & \{3,6^3,6\} & q^2 - q + 1 \\
          C_{11} & \{3^9\}     & q^2 - 2q + 1\\
          C_{12} & \{3,6^4\}   & q^2 + 2q + 1\\
          C_{13} & \{3,12^3\}  & q^2 + 1 \\
          C_{14} & \{9^3\}     & q^2 + q + 1
\end{array}
\]
%Notes: $\eta_j$ is a primitive ${\rm ord}(\eta_j)$th root of unity with  $\alpha_j = \eta_j q$.
%Knowing the $\eta_j$ allows us to compute $N_r$ for all $r \ge 1$ as before.

\end{frame}

\begin{frame}
{\bf Some experimental results}

With $q = 7$, the $s = 1$ (and $s = 2$) codes look like this:

\begin{itemize}
\item $C_{10}$ -- $[43,4,30]$ and $[43,4,31]$ examples (but best known is $d=35$)
\item $C_{11}$ -- $[36,4,23]$ and $[36,4,24]$ examples (but best possible is $28 \le d \le 29$)
\item $C_{12}$ -- $[64,4,51]$ examples (but best possible is $52 \le d \le 53$) (also $s = 2$ with
$[64,10,38]$, but best possible is $41 \le d \le 48$)
\item $C_{13}$ (very rare) -- $[50,4,37]$  (but best known is $d = 42$)
\item $C_{14}$ (rare) -- $[57,4,44]$ (but best known is $d = 47$)
\end{itemize}

\end{frame}

\begin{frame}
{\bf What to make of all this?}

\begin{itemize}
\item Among these, $C_{12}$ cubics are the best for this construction, but still not that
great
\item Plane sections of cubics with Picard number $> 1$ can have up to  $3q + 1$ $\F_q$-points
(Eckardt points as in Amanda Knecht's talk!)  Largest number of $\F_7$-points here is
e.g., $64 - 51 = 13$ ($\Rightarrow$ confirmation of Zarzar's Ansatz)
\item \emph{Why 13?} Hasse-Weil-Serre bound:  The maximum number of $\F_7$-points on
a smooth plane cubic is $1 + 7 + \lfloor 2\sqrt{7}\rfloor = 13$ and attained.  Singular (but irreducible)
plane sections all have either $q = 7$ (``split'' node), $q + 1 = 8$ (cusp), or 
$q + 2 = 9$ (``non-split'' node)
$\F_7$-points.
\item Note: Some $C_{10}$ and $C_{11}$ surfaces have no plane sections with $13$ $\F_7$-points.
\end{itemize}
\end{frame}

\begin{frame}
{\bf A byproduct of this experimentation}

Based on lots of additional experimental evidence for prime powers $q \le 37$,

\begin{conj}
For all $q \ge 5$, $C_{12}$ cubics always have {\bf optimal} cubic plane sections,
i.e. plane sections with the maximum number of $\F_q$-points for a smooth
plane cubic curve.
\end{conj}

Have verified this completely for $q$ up to 13 by ``brute force,'' 
but is there a deeper reason why it should hold?

\vskip10pt
Also would show $s=1$ codes from $C_{12}$ cubics \emph{do not} give any ``new bests'' for larger $q$.

\end{frame}

\begin{frame}
{\bf Some bounds -- {\it sectional genus} of $X$ also matters!}

Notation:  $C(X,s,\F_q) = $ degree $s$ code on a projective surface $X$.

\begin{thm} 
Assume $(\deg X, {\rm char}(\F_q)) = 1$ and Picard number of $X$ $= 1$. 
Writing $d_1 = d(C(X,1,\F_q))$, $g = $
sectional genus,
$$
n - d_1 \le 1 + q + g \lfloor 2\sqrt{q}\rfloor.
$$ 
\end{thm} 

\begin{cor}
In situation of theorem, if $q$ is sufficiently large, then writing $d_s = d(C(X,s,\F_q))$,
$$n - d_s \le s(n - d_1).$$
\end{cor}
\end{frame}

\begin{frame}
{\bf Sectional genus $g = 0$}

\begin{thm}
If $S$ is a smooth abstract surface and $L$ is an ample line bundle with $g(L) = 0$, then 
$(S,L)$ is one of the following:
\begin{itemize}
\item $(\P^2, \mathcal{O}_{\P^2}(r))$, $r = 1,2$.
\item $(Q, \mathcal{O}_Q(1))$
\item a Hirzebruch surface $(F_r, \mathcal{O}_{F_r}(E + sf))$, $s \ge r + 1$.
\end{itemize}
\end{thm}

In other words, few examples, and those are pretty well understood from coding theory 
perspective -- e.g. codes from scrolls (C. Carvalho's talk), toric surface codes.
\end{frame}


\begin{frame}
{\bf Higher sectional genus surfaces not promising}

\begin{itemize}
\item Consider the surface $X_m$ in $\P^3$ given by 
\[
0 = w^m+xy^{m-1}+yz^{m-1}+zx^{m-1}.
\]
Shioda: ${\rm rank}\ NS(X) = 1$ over $\mathbb{C}$ if $m \ge 5$ (and K3 with 
${\rm rank}\ NS(X) = 20$ for $m = 4$).
\item For $m = 4$, reduction of $X_4$ may have no $\F_q$-lines or conics $\Rightarrow$ 
no reducible plane sections.
\item  With $q = 11$ and $s=1$, $C(S_4,1,\F_{11})$ is $[144,4,120]$.  
\item Min. weight codewords $\leftrightarrow$ smooth plane quartics ($g = 3$) 
with $24$ $\F_{11}$-rational points (optimal for $g = 3$ by {\tt manypoints.org}).  
\item $C_{12}$ cubics over $\F_{11}$: all give $[144,4,126]$ codes: $g = 1$ curves over $\F_{11}$ have at most $18$ rational points.  
\item Similarly for $m \ge 5$.
\end{itemize}
\end{frame}

%\begin{frame}
%{\bf Related results}
%
%When $X$ is a cubic surface with $\F_q$-rational lines, one or more can be blown
%down to yield other del Pezzo surfaces.  These also yield some interesting codes, 
%as Hal Schenck and I have discovered:  Over $\F_7$, there are 
%\[
%[50,5,38], [57,5,44]
%\]
%codes constructed this way.  These also equal the best known $d$ for these $n,k$.
%\end{frame}

\section{Additional Examples}

\begin{frame}
{\bf A better sectional genus 1 example}


\begin{itemize}
\item Consider the linear system of cubics in $\P^2$ through a general Frobenius orbit 
${\mathcal O}_3 = \{P, F(P), F^2(P)\}$ ($P \in \P^2(\F_{q^3})$)
\item $\dim = 7$, so defines a rational map $\P^2$ into $\P^6$
\item Image is a degree 6 del Pezzo surface $X$ over $\F_q$, 
\item Blows up the points in ${\mathcal O}_3$ to lines, but
defined over $\F_{q^3}$, not $\F_q$. 
\item $\Rightarrow$ Picard number equal to 2 
\item ${\rm NS}(X)$ is generated by 
classes of proper transforms of conics in $\P^2$ through ${\mathcal O}_3$, and
lines in $\P^2$.
\end{itemize}
\end{frame}

\begin{frame}
{\bf How to determine the Picard number}

\begin{itemize}
\item The zeta function of $X$ has the form
$$Z(X,t) = \frac{[deg\ 0] [deg\ 0]}{[deg\ 1][deg\ 4][deg\ 1]} = \frac{1}{(1 - t)P_2(t)(1 - q^2t)},$$
where $P_2(t) = (1 - qt)\prod_{j=1}^3 (1 - \alpha_j t)$,  with $|\alpha_j| = q$ all $j$.
\item Usual zeta function ``yoga'':
$$|X(\F_{q^r})| = 1 + q^{2r} + q^r + \sum_{j=1}^3 \alpha_j^r = \begin{cases} 1 + q^{2r} + q^r & r \equiv 1,2 \bmod 3\\
1 + q^{2r} + 4q^r & r \equiv 0 \bmod 3\end{cases}$$
\item $\Rightarrow \alpha_j = q, e^{2\pi i/3} q, e^{4\pi i/3} q$.
Tate: the Picard number of $X$ equals $1 + $ the number of $\alpha_j$ 
equal to $q$. 
\end{itemize}

\end{frame}

\begin{frame}
{\bf (More) interesting codes!}

\begin{thm}[also see Couvreur (1)]
$C(X,1,\F_q)$ is a $[q^2 + q + 1, 7, q^2 - q - 1]$
code over $\F_q$. 
\end{thm}
         
%\begin{proof}
%If ${\mathcal O}_3$ is general, there are no $\F_q$-rational lines containing
%all three points in ${\mathcal O}_3$, but there is a 2-parameter family of 
%$\F_q$-rational conics containing those points.  
%Among the codewords of the $C(\P^2, {\mathcal O}_3)$ code, then, we will always
%find some coming from cubics that factor as an $\F_q$-rational conic containing 
%${\mathcal O}_3$ and an $\F_q$-rational line that intersects the conic in two conjugate
%points defined over $\F_{q^2}$.  This gives codewords with $2(q + 1) = 2q + 2$ 
%zeroes.  We claim that this is the largest possible number of zeroes in any nonzero
%codeword, which yields $d = (q^2 + q + 1) - (2q + 2) = q^2 - q - 1$ as in the statement
%to be proved.  The claim follows from a case by case analysis of the possible factorizations
%of a cubic $C$ containing ${\mathcal O}$.  If $C$ is irreducible, the Hasse-Weil-Serre
%bound implies 
%$$|C(\F_q)| \le 1 + q + \lfloor 2\sqrt{q}\rfloor \le (1 + \sqrt{q})^2.$$
%But $(1 + \sqrt{q})^2 < 2 q + 2$
%for all $q \ge 2$.  If $C$ is reducible with a conic containing ${\mathcal O}_3$ as
%one component, then any $\F_q$-rational points in common between the line and the conic
%increase the weight of the evaluation vector, so the words described above with $2q + 2$
%zeroes will have more zeroes.   Finally, there are no reducible cubics with three
%$\F_q$-rational line components in this linear system since if such a line contains
%$P$, then it would have to contain the whole Frobenius orbit ${\mathcal O}_3$.  But there
%are no such lines by our general position hypothesis for ${\mathcal O}_3$.     
%\end{proof}

(Min weight words from reducible cubics: conic through ${\mathcal O}_3$ union a line
meeting the conic in a pair of conjugate $\F_{q^2}$-points)

For $q = 7,8,9$ this equals the best known $d$ for these $n,k$ according to 
Grassl's tables.

\begin{conj}
$C(X,2,\F_q)$ is a $[q^2 + q + 1, 19, q^2 - 3q - 1]$
code over $\F_q$. 
\end{conj}
Would be new best for $q = 7, 9$ and equal best known for $q = 8$.
\end{frame}  





%\begin{frame}
%{\bf Similarly if $m \ge 5$}
%
%The cases $m \ge 5$ are similar.  For instance, $S_5$ over $\F_9$ contains 
%no $\F_9$-rational lines or conics.  The $C(S_5,1,\F_9)$ code has parameters $[91, 4, 71]$ and  
%the nonzero words of minimum weight come from irreducible plane sections ($g = 6$).
%
%\vskip 10pt
%But, best known $d$ in this case is $d = 79$.
%
%\vskip10pt
%(Comment: Although $g = 6$ curves with as many as 32 $\F_9$-rational points are known according
%to {\tt manypoints.org}, none of those are contained in this surface.)
%\end{frame}

\begin{frame}

Thanks for your attention!

\end{frame}

\begin{frame}
{\bf References}

\begin{itemize}
\item[(1)] A. Couvreur, Construction of rational surfaces yielding good codes, {\it Finite Fields Appl.}
{\bf 17} (2011), 424-441.
\item[(2)] J.\ Voloch and M.\ Zarzar, 
Algebraic geometric codes on surfaces,
in ``Arithmetic, geometry, and coding theory", 
{\it S\'emin. Congr.  Soc. Math. France}, {\bf 21} (2010), 211-216.
\item[(3)] M.\ Zarzar, 
Error-correcting codes on low rank surfaces,
{\it Finite Fields Appl.} {\bf 13}  (2007), 727-737.
\end{itemize}
\end{frame}



\end{document}

