In this section we provide some examples of complete NMDS codes in the set of codes constructed above by lifting the elliptic curve \({\mathscr {E}}\) in the case when the base field is large enough.
By Definition 4, the algebraic curve \(\Gamma =\nu _3^2 ({\mathscr {E}})\) provides a complete NMDS code, that is a complete (n; 9, 7)-set of \(\mathrm {PG}(8,q)\), if and only if for any \(Q\in \Sigma \) there exists at least one hyperplane \(\Pi \) of \(\Sigma \) with \(Q\in \Pi \) meeting \(\Gamma \) in 9 points.
Definition 5
We call a point \(Q\in \Sigma \) special for \(\Gamma \) if for all hyperplanes \(\Pi \) of \(\Sigma \) through Q we have \(|\Pi \cap \Gamma |<9\).
For a point Q to be special means that there is a system of cubic curves satisfying one linear constraint such that each element \({\mathscr {C}}\) of this system has intersection multiplicity with \({\mathscr {E}}\) at least 2 in at least one point or meets \({\mathscr {E}}\) in some non-\({\mathbb {F}}_q\)-rational point.
We expect that for large q special points, if they exist at all, are very few. So we propose the following conjecture.
Conjecture 1
Suppose \(q\ge 121\) to be such that \(2,3\not | q\). Then there are no special points for \(\Gamma \).
In order to verify Conjecture 1, we performed some computer searches for some values of q. For \(q \in \{7,11,13\}\) we executed a (non-trivial) exhaustive search. For \(q\ge 121\) we provide an argument showing that there cannot be too many special points, if they exist at all. We leave the solution of the problem and its generalization to a future work.
Search for small q
Recall that any 8 distinct points of \({\mathscr {V}}_3\) are linearly independent; see [9].
For small values of q it is possible to perform an exhaustive search, adopting the following procedure:
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1.
Let \(\Gamma =\nu _3^2 ({\mathscr {E}})\) be the embedding of \({\mathscr {E}}\);
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2.
for any set of 9 points of \(\Gamma \), consider the matrix containing their components; let \({\mathfrak {G}}\) be the list of such matrices having rank 8. In particular, each element of \({\mathfrak {G}}\) corresponds to a hyperplane meeting \(\Gamma \) in 9 points. We call such hyperplanes good.
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3.
For each matrix \(H\in {{\mathfrak {G}}}\), let \(H'\) be a column vector spanning the kernel of H. In particular, we have that a row vector v belongs to the span of the rows of H if and only if \(vH'={\mathbf {0}}\).
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4.
Consider the linear code C with parameters \([|{{\mathfrak {G}}}|,9]\) whose generator matrix G consists of all columns of the form \(H'\) as H varies in \({\mathfrak {G}}\). A point P represented by a vector v can be added to \(\Gamma \) if, and only if, P does not belong to any of the hyperplanes represented by the columns of G; in other words P can be added to \(\Gamma \) if and only if the word PG corresponding to P does not contain any 0-component.
Using the above argument, we can state the following.
Theorem 4.1
The (n;9,7)-set \(\Gamma \) is complete if and only if the code C with generator matrix G constructed above does not contain any word of maximum weight n.
Clearly, it is not restrictive to replace the code C with a code \(C'\) equivalent to C. In particular, if we transform its generator matrix G to row-reduced echelon form, we see that no point with at least a 0 component can give a word of \(C'\) of weight n; this allows to exclude from the search all points whose transforms (under the operations yielding the reduction of C) lie on the coordinate hyperplanes.
We now limit ourselves to the odd order case with q not divisible by 3. Then any elliptic curve \({\mathscr {E}}\) of \(\mathrm {PG}(2,q)\) admits an equation in canonical Weierstrass form
$$\begin{aligned} Y^2 = X^3+aX+b, \end{aligned}$$
with \(a,b\in {\mathbb {F}}_q\) such that \(-16(4a^3+27b^2)\ne 0\); see [12].
Remark
Good hyperplanes correspond to linear systems of cubic curves cutting \({\mathscr {E}}\) in 9 points; by [10, Theorem 43], we see that the number of such hyperplanes is approximately \(\frac{1}{9!}q^7\).
We leave to a future work to determine exactly what sets of 9 distinct points of a given elliptic curve \({\mathscr {E}}\) might arise as intersection divisor with another curve, in other terms to determine what the good hyperplanes are.
Our Conjecture 1 can be restated by saying that the union of all good hyperplanes for \({\mathscr {E}}\) is \(\mathrm {PG}(8,q)\) for q sufficiently large.
We can now apply the aforementioned strategy for all possible values of a, b yielding elliptic curves. This leads to the following.
Theorem 4.2
Suppose \(q\in \{7,11,13\}\). Then, the lifted (n; 9, 7)-set \(\Gamma \) in \(\mathrm {PG}(8,q)\) is complete if and only if \(n=|{\mathscr {E}}|\ge 15\). In particular, for \(q=7\) the lifted set \(\Gamma \) is never complete.
Properties for large q
We now provide an argument to prove that there might not be too many special points. This makes it possible to verify for several values of q that the (n; 9, 7)-set \(\Gamma \) in \(\Sigma =\mathrm {PG}(8,q)\) is complete and gives evidence supporting Conjecture 1.
As in the previous section, the projective plane PG(2, q) is assumed to be of order q odd and not divisible by 3. Furthermore we suppose \(q\ge 121\). Let \(j({\mathscr {E}})\) be the j-invariant of \({\mathscr {E}}\), that is the six cross-ratios of the four tangents from a point of \({\mathscr {E}}\) to other points of \({\mathscr {E}}\). We limit ourselves to the case \(j({\mathscr {E}})\ne 0\), see [8, Theorem 11.15].
We will use the following result which is a direct consequence of [7, Lemma 3.2].
Lemma 4.3
Let \(q\ge 121\) and consider an elliptic cubic \({\mathscr {E}}({\mathbb {F}}_q)\) with \(j({\mathscr {E}})\ne 0\). Then there are at least 7 trisecant \({\mathbb {F}}_q\)-rational lines through any given \({\mathbb {F}}_q\)-rational point.
Up to a change of projective reference, we can assume without loss of generality that the curve \({\mathscr {E}}\) in \(\mathrm {PG}(2,q)\) is met by the reducible cubic \(XYZ=0\) in 9 distinct \({\mathbb {F}}_q\)-rational points.
Lemma 4.4
Under the assumption \(q\ge 121\) any special point \(Q\in \Sigma \) has to be a point \(Q=(0,q_1,q_2,\ldots ,q_9)\in \Sigma \setminus \Gamma \) such that \([q_1,q_3,q_4],[q_4,q_7,q_8] \in {\mathscr {E}}\) and one of the following conditions holds
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\(q_1,q_7=0\); \(q_3,q_4,q_8\ne 0\);
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\(q_1,q_8=0\); \(q_3,q_4,q_7\ne 0\);
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\(q_3,q_7=0\); \(q_1,q_4,q_8\ne 0\);
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\(q_3,q_8=0\); \(q_1,q_4,q_7\ne 0\).
Proof
Let \(Q=(0,q_1,q_2,\ldots ,q_9)\in \Sigma \). If \(Q\in \Gamma \), then Q is not special; indeed, if \(Q\in \Gamma \), then \(Q=\nu _3^2(P)\) with \(P\in {\mathscr {E}}\). Consider a reducible cubic curve \({\mathscr {C}}\) in \(\mathrm {PG}(2,q)\), union of 3 lines \(\ell ,m,r\) with \(P \in \ell \setminus \{m \cup r\} \) and such that \(|(\ell \cup m\cup r)\cap {\mathscr {E}}|=9\). Such a curve if \(|{\mathscr {E}}|>9\) is guaranteed to exist by Lemma 4.3 and it corresponds to a hyperplane of \(\mathrm {PG}(9,q)\) through Q meeting \(\Gamma \) in 9 distinct points. So Q is not special.
Now consider a cubic curve \({\mathscr {C}}\) in \(\mathrm {PG}(2,q)\) with equation of the form
$$\begin{aligned} YZ(\alpha X+\beta Y+\gamma Z)=0, \end{aligned}$$
(3)
and a cubic curve \({\mathscr {C}}'\) with equation of type
$$\begin{aligned} XY(aX+bY+cZ)=0. \end{aligned}$$
(4)
Via the Veronese embedding \(\nu _3^2\), \({\mathscr {C}}\) corresponds to the hyperplane of equation \(\alpha X_4+\beta X_7+\gamma X_8=0\), whereas \({\mathscr {C}}'\) corresponds to the hyperplane \(aX_1+bX_3+cX_4=0\).
For any \(Q\in \Sigma \setminus \Gamma \) write \(P_Q:=[q_4,q_7,q_8]\) and \(P'_Q:=[q_1,q_3,q_4]\in \mathrm {PG}(2,q)\).
If \(P_Q\not \in {{\mathscr {E}}}\), by Lemma 4.3 there are at least 7 lines through \(P_Q\) meeting \({{\mathscr {E}}}\) in 3 distinct points; in particular there is at least one line of equation \(\alpha X+\beta Y+\gamma Z=0\) through \(P_Q\) meeting \({\mathscr {E}}\setminus ([Y=0]\cup [Z=0])\) in 3 distinct points. Consequently the cubic \({\mathscr {C}}:YZ(\alpha X+\beta Y+\gamma Z)=0\) corresponds to a hyperplane \(\Pi \) of \(\mathrm {PG}(9,q)\) through Q, meeting \(\Gamma \) in 9 distinct points and we are done.
If \(P_Q \in {{\mathscr {E}}}\) but \(P'_Q\not \in {\mathscr {E}}\), repeating the same argument starting from a cubic \({\mathscr {C}}'\) with Eq. (4), we see that Q is not special.
Thus, we suppose \(P_Q, P'_Q \in {\mathscr {E}}\) and distinguish several cases:
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1.
If \(q_4=0\), then the cubic \({\mathscr {C}}\) of equation \(XYZ=0\) corresponds to the hyperplane \(X_4=0\) passing through Q with 9 intersections with \(\Gamma \).
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2.
If \(q_4\ne 0\) and \(q_7=q_8=0\), then \(P_Q=[1,0,0]\not \in {{\mathscr {E}}}\), which is excluded.
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3.
If \(q_4\ne 0\) and \(q_1=q_3=0\), then \(P'_Q=[0,0,1]\not \in {{\mathscr {E}}}\), which is excluded.
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4.
Let \(q_4\ne 0\) with \(q_7 \ne 0\) and \(q_8 \ne 0\), then \(P_Q\) is not on \([Y=0]\cup [Z=0]\) in \(\mathrm {PG}(2,q)\). Then, from Lemma 4.3 there are at least 7 lines in \(\mathrm {PG}(2,q)\) through \(P_Q\) which are 3-secants to \({\mathscr {E}}\). Since \({\mathscr {E}}\) has 6 points on the union of the lines \([Y=0]\) and \([Z=0]\), there is at least one line through \(P_Q\) with equation: \(\alpha _1 X+\beta _1 Y+\gamma _1 Z=0\) meeting \({\mathscr {E}}\) in 3 points none of which is on \([Y=0]\) and \([Z=0]\). So, the hyperplane of \(\mathrm {PG}(9,q)\) through Q, corresponding to the cubic \({\mathscr {C}}: YZ(\alpha _1 X+\beta _1 Y+\gamma _1 Z)=0\) meets \(\Gamma \) in 9 points.
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5.
Let \(q_4\ne 0\) , \(q_7\ne 0\) and \(q_8=0\) (or, equivalently, \(q_4\ne 0\), \(q_7=0\) and \( q_8\ne 0\)). Using an argument similar to that of point 4. but starting from a cubic \({\mathscr {C}}'\) through \(P'_Q\) with equation of the form (4), it turns out that if \(q_1\ne 0\) and \(q_3\ne 0\) then the points \(Q(0,q_1,q_2, \ldots , q_7,0,q_9)\) (or \(Q(0,q_1,\ldots ,q_6,0,q_8,q_9)\)) are not special.
Thus, our lemma follows. \(\square \)
Remark
Let \(Q=(0,q_1,\ldots ,q_9)\in \Sigma \) such that Q is not ruled out as special point in Lemma 4.4. For instance, suppose \(q_8=0\) and either \(q_1=0\) or \(q_3=0\) with \([q_1,q_3,q_4]\in {\mathscr {E}}\). So, take \(P(a,0,1)\in \mathrm {PG}(2,q)\setminus {\mathscr {E}}\) and consider a cubic \({\mathscr {C}}\) with equation: \(Y(Y-m_1X+a m_1Z)(Y-m_2 X+a m_2Z)=0\) passing through P meeting \({\mathscr {E}}\) in 9 distinct points. Then, \({\mathscr {C}}\) corresponds to the hyperplane \(\pi : m_1m_2X_1-(m_1+m_2)X_3-2am_1m_2X_4+X_6+a(m_1+m_2)X_7+a^2m_1m_2X_8=0\) which passes through Q if and only if
$$\begin{aligned} m_1m_2q_1- (m_1+m_2)q_3-2am_1m_2q_4+q_6+a(m_1+m_2)q_7=0. \end{aligned}$$
(5)
In particular, if we can determine \(m_1,m_2\) and a such that (5) is satisfied, then the point Q is not special.
A similar argument applies when \(q_7=0\).
Let now \(q \equiv 1\) \(\mod 3\) and \(\omega \) be a root of \(T^2+T+1=0\). Consider a non-singular plane cubic curve \({\mathscr {E}}\) over \({\mathbb {F}}_q\) with canonical equation:
$$\begin{aligned} X^3+Y^3+Z^3-3cXYZ=0, \end{aligned}$$
where \(c \ne \infty , 1,\omega ,\omega ^2\).
If \(c=1+\sqrt{3}\), then the elliptic curve \({\mathscr {E}}\) is harmonic, that is, \(j({\mathscr {E}}) \ne 0\), see [8, Lemma 11.47]. Using Remark 4.2 and the symmetry \(Y\leftrightarrow Z\) of the curve \({\mathscr {E}}\) it is possible to test for the completeness of \(\nu _3^2({\mathscr {E}})\). With the aid of GAP [13], we see that for \(q=121\) we obtain a curve with \(n=144\) rational points, for \(q=157,169\) we obtain curves with \(n=180\) rational points whereas for \(q=179\) we get a curve with \(n=180\) points and in each case the n rational points define a complete NMDS code.