Near-MDS codes from elliptic curves

Abstract

We provide a geometric construction of $[n,9,n-9]_q$ near-MDS codes arising from elliptic curves with n ${\mathbb {F}}_q$-rational points. Furthermore, we show that in some cases these codes cannot be extended to longer near-MDS codes.

1 Introduction

Maximum distance separable (for short MDS) codes are the best linear $[n,k,d]_{q}$ codes as they meet the Singleton bound, that is, $n=d+k-1$. The non-negative integer $s(\mathbf{C}):= n-k+1-d$ is said to be the Singleton defect of the code $\mathbf{C}$. Thus, the Singleton defect of an MDS code is zero.

A linear code $\mathbf{C}$ is defined to be a near-MDS (for short NMDS) code if $s(\mathbf{C})=s(\mathbf{C}^{\bot })=1$ where $\mathbf{C}^{\bot }$ is the dual code of $\mathbf{C}$. Hence, a NMDS [n, k] code has minimum distance $n-k$.

NMDS codes were introduced by Dodunekov and Landjev [4] with the aim of constructing good linear codes by slightly weakening the restrictions in the definition of an MDS code. NMDS codes have similar properties to MDS codes. Some non-binary linear codes such as the ternary Golay codes, the quaternary quadratic residue $[11,6,5]_{4}$-code, and the quaternary extended quadratic residue $[12,6,6]_{4}$-code are notable examples of NMDS codes; see [11].

The geometrical counterpart of an NMDS code is an n-track in a Galois space which is a set of n points in an N-dimensional Galois space such that every N of them are linearly independent but some $N+1$ of them, see [3]. If every $N+2$ points of the n-track generate the whole space then the $n\times (N+1)$ matrix whose columns are homogeneous coordinates of the n-track points is a generator matrix of an NMDS code. The n-track is complete, i.e. maximal with respect to set theoretical inclusion, if and only if the code is not extendable.

Let $N_q$ denote the maximum number of ${\mathbb {F}}_q$-rational points on an elliptic curve defined over ${\mathbb {F}}_q$; it is well-known that, by Hasse theorem, $|N_q-(q+1)|\le 2\sqrt{q}$.

NMDS codes of length up to $N_q$ may be constructed from elliptic curves. An interesting question is whether there exist NMDS codes of length greater than $N_q$. Constructions of NMDS codes from elliptic curves are found in [1, 2, 7] where results both from combinatorics and algebraic geometry are used.

Here we provide a geometric construction of 9 dimensional NMDS codes using an algebraic curve of order 9 in $\mathrm {PG}(9, q)$ which arises from a non-singular cubic curve ${\mathscr {E}}: f(X,Y,Z)=0$ of $\mathrm {PG}(2, q)$ via the (modified) Veronese embedding:

$$\begin{aligned}&\nu _{3}^2 :(X{:}Y{:}Z)\mapsto \nonumber \\&\big (f(X,Y,Z)\,{:}\,X^2Y\,{:}\,X^2Z\,{:}\,XY^2\,{:}\,XYZ\,{:}\,XZ^{2}\,{:}Y^3\,{:}\,Y^2Z\,{:}\,YZ^{2}\,{:}\,Z^{3}\big ). \end{aligned}$$

(1)

We also show that certain codes from elliptic curves are not extendible to longer NMDS codes. The proof depends on some results on the number of ${\mathbb {F}}_q$-rational lines through a given point P that meet a plane elliptic curve in exactly three ${\mathbb {F}}_q$-rational points and on some computations carried out with the aid of GAP [13].

2 Preliminaries

The following definitions of an NMDS code of length n and dimension k over a finite field ${\mathbb {F}}_{q}$ are equivalent to that given in the Introduction; see [5].

Definition 1

A linear [n, k] code over ${\mathbb {F}}_{q}$ is NMDS if any of its generator matrices, say G, satisfies the following conditions:

• (i) any $k-1$ columns of G are linearly independent;

• (ii) G contains k linearly dependent columns;

• (iii) any $k+1$ columns of G have full rank.

Definition 2

A linear [n, k] code over ${\mathbb {F}}_{q}$ is NMDS if any of its parity check matrices, say H, satisfies the following conditions:

• (i) any $n-k-1$ columns of H are linearly independent;

• (ii) H contains $n-k$ linearly dependent columns;

• (iii) any $n-k+1$ columns of H have full rank.

From a geometric point of view, a NMDS [n, k] code ${\mathbf {C}}$ over ${\mathbb {F}}_{q}$ can be regarded as a projective system (i.e. a distinguished point set) ${\mathbf {C}}$ in a projective space $\mathrm {PG}(k-1,q)$; see [14] for more details.

Definition 3

A subset ${\mathbf {C}} \subseteq \mathrm {PG}(k-1,q)$ is an $(n; k, k-2)$-set in $\mathrm {PG}(k-1,{\mathbb {F}}_{q})$ if it satisfies the following conditions:

• (i) every $k-1$ points in ${\mathbf {C}}$ span a hyperplane of $\mathrm {PG}(k-1,q)$;

• (ii) there exists a hyperplane of $\mathrm {PG}(k-1,q)$ containing exactly k points of ${\mathbf {C}}$;

• (iii) every $k+1$ points of ${\mathbf {C}}$ generate the whole $\mathrm {PG}(k-1,q)$.

Definition 4

An $(n; k, k-2)$-set in $\mathrm {PG}(k-1,{\mathbb {F}}_{q})$ is complete if it is maximal with respect to set-theoretical inclusion.

Thus, in this setting, an NMDS [n, k] code over ${\mathbb {F}}_{q}$ is an $(n; k, k-2)$-set in $\mathrm {PG}(k-1,{\mathbb {F}}_{q})$.

Given an integer $\nu \ge 1$ and a prime power $q=p^{h}$, consider the set ${\mathfrak {C}}^{\nu }$ of all the curves of degree $\nu$ contained in the projective plane $\mathrm {PG}(2,q)$ over a finite field ${\mathbb {F}}_{q}$. Since any curve ${\mathscr {C}}\in {\mathfrak {C}}^{\nu }$ is uniquely determined by $m+1=\left( {\begin{array}{c}\nu +2\\ 2\end{array}}\right)$ parameters in ${\mathbb {F}}_{q}$, that is, the coefficients of its equation

$$\begin{aligned}&a_{0}Z^{\nu }+(a_{1}X+a_{2}Y)Z^{\nu -1}+(a_{3}X^{2}+a_{4}XY+a_{5}Y^{2})Z^{\nu -2}+\cdots \\&+(a_{m-\nu }X^{\nu }+a_{m-\nu +1}X^{\nu -1}Y+\cdots +a_{m-1}XY^{\nu -1}+a_{m}Y^{\nu })=0, \end{aligned}$$

and the curve is unchanged if these parameters are multiplied by a common factor, then ${\mathfrak {C}}^{\nu }$ can be regarded as a projective space $\mathrm {PG}(m,q)$ with homogeneous coordinates $(a_{0}{:}a_{1}{:}\cdots {:}a_{m})$. We may also denote a curve ${\mathscr {C}}$ by using its defining polynomial.

The following result—which is an implicit formulation of the famous Cayley-Bacharach theorem—will be useful later; see [6].

Theorem 2.1

Let ${\mathscr {E}}$ and ${\mathscr {C}}$ be two distinct cubic curves meeting in a set ${\mathscr {S}}$ consisting of 9 points (counted with multiplicities). If ${\mathscr {D}}\subset \mathrm {PG}(2,q)$ is any cubic curve containing all but one point of ${\mathscr {S}}$, then ${\mathscr {C}}\cap {\mathscr {D}}={\mathscr {S}}$.

3 Lifting point sets

The space ${\mathfrak {C}}^{3}$ consisting of all the cubics in $\mathrm {PG}(2,q)$ has projective dimension 9, hence 10 independent cubic curves are required to generate it. Let ${\mathscr {E}}$ be a non-singular cubic curve of equation $f(X,Y,Z)=0$ over ${\mathbb {F}}_q$. A suitable basis ${\mathcal {B}}$ for ${\mathfrak {C}}^{3}$, containing ${\mathscr {E}}$, can be written by using the following polynomials:

$$\begin{aligned} {\mathcal {B}}=\{f(X,Y,Z),\ X^{2}Y ,\ X^2Z,\ XY^2,\ XYZ,\ XZ^2,\ Y^3,\ Y^{2}Z,\ YZ^{2},\ Z^{3}\}, \end{aligned}$$

where f(X, Y, Z) is required to contain the term $X^{3}$. In fact, the defining polynomial of any cubic curve would be suitable as first element of the basis ${\mathcal {B}}$, as long as it contains the monomial $X^3$; nevertheless, the choice of an elliptic curve is motivated by the fact that, unlike the case of genus 0, the number of ${\mathbb {F}}_q$-rational points of a carefully chosen elliptic curve is not necessarily limited to $q+1$.

We consider the following embedding of the points of $\mathrm {PG}(2,q)$ onto $\mathrm {PG}(9,q)$ with projective coordinates $(X_{0}{:}X_{1}{:}X_{2}{:}X_{3}{:}X_{4}{:}X_{5}{:}X_{6}{:}X_{7}{:}X_{8}{:}X_{9})$ by means of the mapping $\nu _3^2:\mathrm {PG}(2,q)\rightarrow \mathrm {PG}(9,q)$  (1) which is a Veronese embedding of degree 3. Let ${\mathscr {V}}_3$ be the image of $\nu _3^2$; clearly ${\mathscr {V}}_3$ is (projective equivalent to) the cubic Veronese surface.

More in detail, the points of the curve ${\mathscr {E}}$ are mapped onto a curve $\Gamma$ of $\mathrm {PG}(9,q)$ with the same number n of ${\mathbb {F}}_q$-rational points as ${\mathscr {E}}$. Also, $\Gamma$ is the complete intersection of ${\mathscr {V}}_3$ with the hyperplane $\Sigma \cong \mathrm {PG}(8,q)$ of equation $X_0=0$. Since for every cubic curve ${\mathscr {C}}$ of equation $g(X,Y,Z)=0$ in $\mathrm {PG}(2,q)$, the defining polynomial is a linear combination of the elements of ${\mathcal {B}}$, that is,

$$\begin{aligned} g(X,Y,Z)&=\lambda _{0}f(X,Y,Z)+\lambda _{1}Y^{3}+\lambda _{2}XZ^{2}+\lambda _{3}YZ^{2}+\lambda _{4}X^{2}Z \\&\quad +\lambda _{5}Y^{2}Z+\lambda _{6}XYZ+\lambda _{7}X^{2}Y+\lambda _{8}XY^{2}+\lambda _{9}Z^{3}, \end{aligned}$$

it turns out that $\nu _3^2({\mathscr {C}})$ is the complete intersection of ${\mathscr {V}}_3$ with the hyperplane $\Pi \subset \mathrm {PG}(9,q)$ of equation

$$\begin{aligned} \sum _{i=0}^{9}\lambda _{i}X_{i}=0, \end{aligned}$$

(2)

which is distinct from $\Sigma$. Thus, every cubic curve ${\mathscr {C}}: g(X,Y,Z)=0$ of $\mathrm {PG}(2, q)$ corresponds to a hyperplane of Eq. (2). Back to $\mathrm {PG}(2,q)$, the set $(\nu _3^2)^{-1}(\Pi \cap {\mathscr {V}}_3)$ corresponds to a unique cubic curve ${\mathscr {C}}$ distinct from ${\mathscr {E}}$, and, clearly, $(\nu _3^2)^{-1}(\Pi \cap \Gamma )$ corresponds to ${\mathscr {C}}\cap {\mathscr {E}}$.

Theorem 3.1

Suppose that ${\mathscr {E}}$ has $n \ge 9$ points. Then the point set $\Gamma$ is an (n; 9, 7)-set in $\Sigma =\mathrm {PG}(8,q)$.

Proof

To prove the theorem it suffices to consider the mutual position of cubic curves in $\mathrm {PG}(2,q)$.

• (i) Take eight distinct points $P_{1},\ldots ,P_{8}\in \Gamma$ and consider the corresponding distinct points $Q_{1},\ldots , Q_{8}\in {\mathscr {E}}$, with $Q_{i}=(\nu _3^2)^{-1}(P_{i})$. Suppose that there is a t-dimensional net with $t\ge 2$, say ${\mathscr {F}}$, consisting of cubics through $Q_{1},\ldots ,Q_{8}$. Then, from Theorem 2.1 there is a ninth point $Q_{9}\in {\mathscr {E}}$ such that the points $Q_{1},\ldots ,Q_{9}$ are in the support of ${\mathscr {F}}$. This implies that every further point $Q_{10}\in {\mathscr {E}}\setminus \{Q_{1},\ldots ,Q_{9}\}$ yields a $(t-1)$-dimensional net consisting of cubics through $Q_{1},\ldots ,Q_{9}$ which are distinct from ${\mathscr {E}}$ and have ten points in common with it, contradicting Bézout’s theorem. Hence, ${\mathscr {F}}$ must be a pencil of cubic curves in $\mathrm {PG}(2,q)$ including ${\mathscr {E}}$ and passing through $Q_{1},\ldots ,Q_{8}$. Back to $\mathrm {PG}(9,q)$, we observe that ${\mathscr {F}}$ corresponds to a pencil of hyperplanes of $\mathrm {PG}(9,q)$ which meet in a unique 7-dimensional subspace $\Delta$ such that $\{P_{1},\ldots ,P_{8}\}\subset (\Gamma \cap \Delta )$, that is, $P_{1}$, …, $P_{8}$, generate the hyperplane $\Delta$ of $\Sigma$.

• (ii) From Theorem 2.1, there is a further point $Q_{9}\in \mathrm {PG}(2,q)$ which belongs to the intersection of ${\mathscr {E}}$ and all the other cubics of the above pencil ${\mathscr {F}}$. This proves that the previous subspace $\Delta$ meets $\Gamma$ in $P_{1}$,…,$P_{8}$, $P_{9}=\nu _3^2 (Q_{9})$.

• (iii) Let $\Pi$ be a hyperplane of $\mathrm {PG}(9,q)$ different from $\Sigma$. Put ${\mathscr {C}}=(\nu _3^2)^{-1}(\Pi )$. From Bézout’s theorem we know that $\vert {\mathscr {E}}\cap {\mathscr {C}}\vert \le 9$, therefore any hyperplane of $\mathrm {PG}(9,q)$ has at most 9 points in common with $\Gamma$. Hence, $\Gamma$ is a curve of order 9, therefore 10 points of $\Gamma$ generate the whole $\Sigma$.

The claim follows. $\square$

Remark

The code associated to $\Gamma$ can also be interpreted as an AG-code, see [14]. Indeed, Theorem 3.1 is a consequence of [14, Theorem 4.4.19]. However, our proof does not use the Riemmman-Roch Theorem.

4 Some complete NMDS codes

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.

4.1 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:

• 1. Let $\Gamma =\nu _3^2 ({\mathscr {E}})$ be the embedding of ${\mathscr {E}}$;

• 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.

• 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 $vH'={\mathbf {0}}$.

• 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.

4.2 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

• $q_1,q_7=0$; $q_3,q_4,q_8\ne 0$;

• $q_1,q_8=0$; $q_3,q_4,q_7\ne 0$;

• $q_3,q_7=0$; $q_1,q_4,q_8\ne 0$;

• $q_3,q_8=0$; $q_1,q_4,q_7\ne 0$.

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:

• 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 $\Gamma$.

• 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.

• 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.

• 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.

• 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.