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\begin{document}

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\title{Gr\"obner bases and minimum distance of affine varieties codes\thanks{The author was supported in part by  CNPq grants  302280/2011-1 and  470416/2011-4.}
}


\author{C.\  CARVALHO\thanks{cicero@ufu.br}\, , Faculdade de Matem\'atica, Universidade Federal de Uberl\^andia,  Av.\ J.\ N.\ \'Avila 2121, 38.408-902 - Uberl\^andia - MG, Brazil.}

\criartitulo



\runningheads {Carvalho}{Gr\"obner bases and minimum distance of affine varieties codes}

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\begin{abstract}
{\bf Abstract}.
We present a method to estimate the minimum distance of  affine varieties codes. Our technique uses properties of the footprint of an ideal obtained by  enlarging the defining ideal of the variety, and may be applied also to codes which do not come from the so-called weight domains.
 
{\bf Keywords.} Affine varieties codes, Gr\"obner bases, footprint of an ideal, minimum distance
\end{abstract}

\noindent
\small{\textbf{AMS Classification:} 13P10, 13F20, 94B27 

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\newsec{Introduction}

Since the appearance of the geometric Goppa codes in the eighties, many papers have dealt with improvements on the lower bound for the minimum distance. One of the most successful methods for this improvement was obtained by Feng and Rao (see \cite{fr1} and \cite{fr2}). Many related bounds appeared after their work, one of them being a bound derived by Andersen and Geil in \cite{and-geil}. In that paper the authors first derive a general approach to obtain a bound for the minimum distance (actually, for the generalized Hamming weights) of a linear code, and then show how to apply their method to codes defined from weight domains. 
A weight domain is an $\F$-algebra, where $\F$ is a field, which admits a function to $\mathbb{N}_0 \cup \{-\infty\}$ satisfying certain properties, which makes the domain suitable to be used for defining codes, when $\F$ is a finite field. They were introduced in \cite{h-vl-p} by T.\ H\o holdt,  J.\ H.\ van Lint and R.\ Pellikaan  in order to present an alternative construction for geometric Goppa codes with simple tools from commutative algebra. In the present work we show how to apply Andersen and Geil's general approach to affine varieties codes. Similarly to codes obtained from weight domains, these are evaluation codes obtained from the ring of regular functions of an affine variety but weight functions play no role in this theory. Since the algebras which appear in the weight function theory are the ring of regular functions of certain type of  varieties (see \cite{g-p}) our result applies to a more comprehensive class of rings (see Example \ref{ex2}). Thus, distinctly from recent works (see e.g.\ \cite{geil1} and \cite{geil2}) we do not need concepts like ``well-behaving basis'' or ``one-way well behaving basis''.   An important set of data to obtain the bound is the set of indexes where there is a ``dimension jump'' in a sequence of nested vector spaces. While in \cite{and-geil} there are several results on such set for the case of codes from weight domains, and in particular, one-point geometric Goppa codes, here we show that this set may be read directly from the footprint of an ideal obtained by enlarging the defining ideal of the curve.

In the next section we recall Andersen and Geil's approach to obtain a bound for the minimum distance of a linear code, we introduce the affine varieties codes and recall the definition and some properties of the footprint of an ideal, then we prove our main result. Following that, we  present some examples to illustrate our method, including codes obtained from an algebra which does not admit an weight function.



\newsec{Main  result}

Let $\F_q$ be a finite field with $q$ elements, $n$ a positive integer and for $\mathbf{a} := (a_1, \ldots, a_n)$, $\mathbf{b} := (b_1, \ldots, b_n) \in \F^n$ define $\mathbf{a} * \mathbf{b} := (a_1 b_1, \ldots, a_n b_n)$. Let $C$ be a vector subspace of $\F_q^n$. The idea of Andersen and Geil for finding  a lower bound for the minimum distance of $C$ stems from the fact that  if $\mathbf{c} \in C$ and $\{\mathbf{b}_1, \ldots, \mathbf{b}_n\} =:\mathbf{B}$ is a basis for $\F^n$ then the subspace $\mathbf{c}*\mathbf{B}$ generated by $\{\mathbf{c} * \mathbf{b}_1, \ldots, \mathbf{c}*\mathbf{b}_n\}$ has dimension equal to the weight of $\mathbf{c}$. Thus we have the following result.

\begin{lemma}\label{2.1} The minimum distance $d(C)$ is equal to  $\min \{ \dim \mathbf{c}*\mathbf{B} \; ; \mathbf{c} \in C \setminus\{0\}\; \}$.
\end{lemma}

We will use this result to estimate the minimum distance of the so-called affine varieties codes, which were introduced by J. Fitzgerald and R. F. Lax in \cite{f-lax}. Let $I \subset F_q[X_1, \ldots, X_m]$ be an ideal, let $V_{\F_q}(I) = \{P_1, \ldots, P_n\}$ be the associated variety of $\F_q$-rational points and set $R := F_q[X_1, \ldots, X_m]/I$. Consider the evaluation morphism $\varphi: R \rightarrow \F_q^n$ given by $f + I \mapsto (f(P_1),\ldots, f(P_n))$ and let $L$ be an $\F_q$-vector subspace of $R$. 

\begin{definition} The {\em affine variety code} $C(L)$ is the image $\varphi(L)$.
\end{definition}

We observe that as an $\F_q$-vector space $R$ may not have finite dimension. A useful way of finding a basis for $R$ is by means of the so-called footprint an ideal.

\begin{definition} Assume that $F_q[X_1, \ldots, X_m]$ is endowed with a monomial order $\meq$. The {\em footprint} of $I$ (with respect to $\meq$), denoted by $\Delta(I)$, is the set of monomials which are not leading monomials of any polynomial in $I$.
\end{definition}

Let $\Delta(I) = \{M_\lambda \; | \; \lambda \in \Lambda \}$ be the footprint of $I$ with respect to a monomial order $\meq$; one of the main properties of $\Delta(I)$ is that $\{M_\lambda + I \; | \; \lambda \in \Lambda \}$ is a basis for $R$ as an $\F_q$-vector space (see e.g. \cite[Prop. 4, $\S$ 3, Ch. 5]{iva}), and we observe that it is a basis which already carries an order. Thus, for each $\lambda \in \Lambda$ we   consider the $\F_q$-subspace  $L_\lambda \subset R$ which is  generated by all monomials in $\Delta(I)$ which are less or equal than $M_\lambda$. Clearly, if $M_\s \meq M_\lambda$ then $L_\s \subseteq L_\lambda$, so that $C(L_\s) \subseteq C(L_\lambda)$. The next result shows for which values of $\lambda$ we get $C(L_\s) \subsetneq C(L_\lambda)$. 

\begin{theorem} \label{main} Let $I_q := I + (X_1^q - X_1, \ldots, X_m^q - X_m)$. Then $\dim C(L_\lambda) > \dim C(L_\s)$ (with $M_\lambda \succ M_\s$) if and only if $M_\lambda \in \Delta(I_q)$.
\end{theorem} 
\begin{proof} Observe initially that since $I \subset I_q$ then $\Delta(I_q) \subset \Delta(I)$, so that the claim makes sense. Denote by $\overline{\F_q}$ an algebraic closure of $\F_q$, clearly we have
$V(I) = V(I_q)$ and denoting by $V_{\overline{\F_q}}(I_q) \subset \overline{\F_q}^m$ the variety of $I_q$ as an  ideal of $\overline{\F_q}[X_1, \ldots, X_m]$ we also get $V(I_q) = V_{\overline{\F_q}}(I_q)$ (considering the natural inclusion $ {\F_q}^m \subset \overline{\F_q}^m$). From Seidenberg's Lemma 92 (see \cite{sei} or \cite[Lemma 8.13]{becker}) we get that $I_q$ is a radical ideal, so from \cite[Thm. 8.32]{becker} we get that $R/I_q$ is an $\F_q$-vector space of finite dimension $\# V_{\overline{\F_q}}(I_q)$, and since the classes of the monomials in $\Delta(I_q)$ form a basis for $R/I_q$ we get $\# \Delta(I_q) = n$. We will prove now that if $\lambda \in \Delta(I_q)$ then  $\dim C(L_\lambda) > \dim C(L_\s)$ for any $M_\s \prec M_\lambda$. Assume that it is not the case, 
so there exists $\s$ with $M_\s \meqn M_\lambda$ such that $ C(L_\lambda) = C(L_\s)$. In particular, there exists a nonzero finite linear combination $\sum_{M_{s'} \meq M_s} a_{s'} M_{s'} \in L_\s$ such that  $(\sum_{M_{s'} \meq M_s} a_{s´} M_{s'} ) (P_i) = M_\lambda(P_i)$ for all $i = 1, \ldots, n$. 
From Hilbert's Nullstellensatz (see e.g.\ \cite[Thm.\ 2, $\S 1$, Ch.\ 4]{iva}) we get that
the polynomial $M_\lambda - \sum_{M_{s'} \meq M_s} a_{s´} M_{s'}$ is in $\sqrt{I_q} = I_q$ and  a fortiori $M_\lambda \notin \Delta(I_q)$, a contradiction. This completes the proof of the ``if'' assertion, for the ``only if'' part observe that the dimension of the spaces $\dim C(L_\lambda)$ may jump from 1 to $n$ at most $n - 1$ times, but we just proved that it will jump $n-1$ times, so every jump must correspond to an element of $\Delta(I_q)$.
\end{proof}

Note that, incidentally, we have proved that $\varphi$ is surjective.

Let $\Delta(I_q) = \{ M_{\lambda_1}, \ldots,  M_{\lambda_n} \}$, where 
$M_{\lambda_i} \meqn M_{\lambda_{i + 1}}$ for $i = 1, \ldots, n - 1$;
we know then that $\{ \mathbf{b}_1 := \varphi(M_{\lambda_1}), \ldots,  \mathbf{b}_n := \varphi(M_{\lambda_n}) \}$ is a basis for $\F_q^n$. 
Let $\mathbf{c} = \sum_{j = 1}^t  a_j \mathbf{b}_j$, where $t \in \{1, \ldots, n\}$, $a_j \in \F_q$ for all $j \in \{1, \ldots, t\}$ and $a_t \neq 0$; to apply Lemma \ref{2.1} it would be nice to know for a given value of $i \in \{1, \ldots, n\}$ which is 
the least value $\ell(i)$ such that $\mathbf{c}*\mathbf{b}_i \in C(L_{\ell(i)})$, since the number of distinct elements in $\{\ell(1), \ldots, \ell(n)\}$ is equal to $\dim \mathbf{c}*\mathbf{B}$. Because of the nature of the basis we chose, such $\ell(i)$ may usually be determined by calculating the remainder in the division of $M_{\lambda_t} M_{\lambda_i}$ by a Gr\"obner basis with respect to $\meq$, as we show in the examples below. In many cases it happens that this remainder is the product itself, and this may be checked by simple inspection of the basis. 



\newsec{Examples}
\begin{example} For the first example, we take the hermitian curve given by $Y^3 + Y - X^4 = 0$ and defined over $\F_9$. Codes over this curve has been studied extensively, and the minimum distance of one point geometric Goppa codes has been determined by Stichtenoth (\cite{sti}) and Yang and Kumar (\cite{y-k}). Building on the experience of those who have dealt with these codes we choose a degree-lexicographic order for $\F_9[X,Y]$ by stating that $X^a Y^b \meq X^{a'} Y^{b'}$ if and only if $3 a + 4 b \leq 3 a' + 4 b'$, and if ``=" holds then   $X^a Y^b <_{\rm{lex}} X^{a'} Y^{b'}$ (with $Y <_{\rm{lex}} X$). These weights come from the pole orders of the rational functions $x = X/Z$ and $y = Y/Z$ at the point at infinity $P_\infty := (0:1:0)$, which is their only pole. Using $\cocoa$ (\cite{coc}) or Macaulay2 (\cite{mac})  we may calculate a Gr\"obner basis for the ideal $I_9 := (Y^3 + Y - X^4, Y^9 -Y, X^9 - X)$ with respect to $\meq$, and from that we may find the footprint of $I_9$ (w.r.t. $\meq$) as being
$\Delta(I_9) = \{X^a Y^b \;| \; 0 \leq a \leq 3, 0 \leq b \leq 5\} \cup \{Y^6, Y^7, Y^8\}$. The curve has 27 rational points (in the affine plane), which is the same number of elements in the footprint, as expected from the proof of theorem \ref{main}. Let's assume that the code $C$ is generated by the evaluation of (the class of) the first 5 elements of the basis, namely, $\{1, X, Y, X^2, XY\}$ in the 27 points (so the code will have dimension 5, since we know from the above theorem that $\{\varphi(1 + I), \varphi(X + I), \varphi(Y + I), \varphi(X^2 + I), \varphi(XY + I)\}$ is a linearly independent set in $\F_9^{27}$). Let $f = a_1 + a_2 X + a_3 Y + a_4 X^2 + a_5 XY$ with $a_i \in \F_9$ for $i = 1, \ldots, 5$ and $a_5 \neq 0$, set $\varphi(f + I) = \mathbf{c}$. Let $N \in \{1, X, Y, X^2, XY\}$ and $M \in \mathcal{M} := \{X^a Y^b \;| \; 0 \leq a \leq 1, 0 \leq b \leq 4\}$, it's easy to check that $N M \in \Delta(I_9)$ so from the property of the footprint and the theorem above we may conclude that $\mathbf{c}*\varphi(M) = \varphi(f.M)$ is in the subspace $L_{XY.M} \subset \F_9^{27} $ generated by $\varphi(T)$, where $T \in\Delta(I_9)$ and $T \meq XY. M$. Actually, we may conclude more, namely, that if $g = b_1 + b_2 X + b_3 Y + b_4 X^2 + b_5 XY$ is a nonzero polynomial,  with $b_i \in \F_9$ for $i = 1, \ldots, 5$, then the space generated by $\varphi(g)*\varphi(M)$, with $M \in \mathcal{M}$ has dimension $10 = \# \mathcal{M}$. Now, if we want to determine in which space lies, say, $\mathbf{c}*\varphi(X^2)$ we will have to compute the remainder of $X^2.X^2$ in the division by the elements of the Gr\"obner basis in order to write the class of $X^2.X^2$ as a linear combination of $\{ M + I \; | \; M \in \Delta(I_9) \}$. We may do that with the already mentioned computer algebra systems and  then we conclude that the space generated by $\mathbf{c}*\varphi(M)$, with $M \in \mathcal{M} \cup \{X^2\}$ has dimension 11. Proceeding like this, we find that the space generated by $\mathbf{c}*\varphi(M)$, with $M \in \Delta(I_9) \setminus \{Y^5, Y^6,Y^7,Y^8, X Y^{15}, X^2 Y^{15}, X^3 Y^{15} \}$ has dimension 20. Doing the same with $g = b_1 + b_2 X + b_3 Y + b_4 X^2 + b_5 XY \neq 0$ with $b_i \in \F_9$ for $i = 1, \ldots, 5$, we find that 20 is a lower bound for the minimum distance, and in fact it is the actual bound, which we may check by observing that the code corresponds to the geometric Goppa code generated by a basis of $L(7 P_\infty)$, see \cite{sti}.
\end{example}


\begin{example} \label{ex2} In this example we deal with the affine curve defined over $F_9$ by the equation $X^6 Y^4 + X^8 + 1 = 0$. Observe that the closure of the curve in $\mathbb{P}^2(\mathbb{F}_9)$ has two nonsingular points, namely, $P_1 := (0:1:0)$ and $P_2 := (1:0:0)$ so $R = \mathbb{F}_9[X,Y]/(X^6 Y^4 + X^8 + 1)$ is {\em not} an weight domain (see \cite{mat}, see also \cite{c-s} for results on  codes defined by means of near weight domains, which include the ring $R$). The pole divisor of the functions $x$ and $y$ in the function field of the curve are respectively, $\div_\infty(x) = 2 P_1 + 2 P_2$ and  $\div_\infty(y) = 5 P_1 + 5 P_2$ (calculations done with KASH/KANT - \cite{kash}) so we choose a degree-lexicographic order for $\F_9[X,Y]$ by stating that $X^a Y^b \meq X^{a'} Y^{b'}$ if and only if $2 a + 5 b \leq 2 a' + 5 b'$, and if ``=" holds then   $X^a Y^b <_{\rm{lex}} X^{a'} Y^{b'}$ (with $Y <_{\rm{lex}} X$). We now calculate a Gr\"obner basis for $I_9 = (X^6 Y^4 + X^8 + 1 , Y^9 - Y, X^9 - X)$ and find that the footprint is $\Delta(I_9) = \{ X^a Y^b \; | \; 0 \leq a \leq 3, 0 \leq b \leq 3 \}$, and we conclude that there are 16 rational points in the affine curve. Let $C$ be the code generated by evaluating the classes $\{1 + I, X + I , X^2 + I, Y + I, X^3 + I, XY + I \}$ at the rational points in the affine plane (hence, from the above theorem we know that this code has dimension 6). Observe that this is the geometric Goppa code generated by a base of  $L(7 P_1 + 7 P_2)$, or in other words, the geometric Goppa code associated with the divisors $G = 7 P_1 + 7P_2$ and $D$, where $D$ is the sum of all rational points. Let $f = a_1 + a_2 X + a_3 X^2 + a_4 Y + a_5 X^3 + a_6 XY$, with $a_1,\ldots, a_6 \in \F_9$ and let $j \in \{1, \ldots, 6\}$ be greatest index for which $a_j \neq 0$. Denoting by $\mathbf{B}$ the (ordered) basis  $\{ \varphi(X^a Y^b + I) \; | \; 0 \leq a \leq 3, 0 \leq b \leq 3 \}$ of $\mathbb{F}_9^{16}$ and proceeding as in the example above we see that $\varphi(F)*\mathbb{B}$ has dimension at least  9 (respectively, 4, 12, 8, 12, 16) if $j = 6$ (respectively, 5, 4, 3, 2, 1), so our bound for the minimum distance is 4, and one may check that this is the actual bound. We also see that if we discard $X^3$ and consider the code generated by evaluating the classes $\{1 + I, X + I , X^2 + I, Y + I,  XY + I \}$ at the rational points, then the bound for the minimum distance is now 8, and again one may check that this is the actual bound.
\end{example}
\vspace{3ex}

\begin{abstract}
{\bf Resumo}.
Nesse trabalho apresentamos um método para estimar a distância mí\-ni\-ma de códigos de variedades afins. Nossa técnica usa propriedades da pegada de um ideal obtido através do aumento do ideal de definição da variedade em questão, e também pode ser aplicada a códigos de que não são produzidos utilizando-se domínios-pesos.
\end{abstract}


%\section{Conclusions} 
%We have proved that the footprint of an extended ideal $I_q$ yields important parameters to determine a bound for the minimum distance of 
%a code defined over the affine variety determined by $I$, in the case where $I$ is radical. In the talk we would like to also mention how to apply this result to codes coming from algebras that admit a set of near weight functions (as the one in Example 3.2), and present applications to the construction of improved codes (as done in the end of Example 3.2). 


%defined by $L_\lambda := \langle \{ M_\s \in \Delta(I) \; | \; M_s \meq M_\lambda \} \rangle$, i.e $L_\lambda$ is the vector space

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%% MUDAR NA VERSAO FINAL

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\end{document}

