Near-MDS codes from elliptic curves

We provide a geometric construction of [n,9,n-9]q\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$[n,9,n-9]_q$$\end{document} near-MDS codes arising from elliptic curves with nFq\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb {F}}_q$$\end{document}-rational points. Furthermore, we show that in some cases these codes cannot be extended to longer near-MDS codes.


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(C) := n − k + 1 − d is said to be the Singleton defect of the code C. Thus, the Singleton defect of an MDS code is zero.
A linear code C is defined to be a near-MDS (for short NMDS) code if s(C) = s(C ⊥ ) = 1 where C ⊥ is the dual code of 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 [13].
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×(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 F q -rational points on an elliptic curve defined over F q ; it is well-known that, by Hasse theorem, 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 [1,2,8] 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 PG(9, q) which arises from a nonsingular cubic curve E : f (X, Y, Z) = 0 of PG(2, q) via the (modified) Veronese embedding: 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 F q -rational lines through a given point P that meet a plane elliptic curve in exactly three F q -rational points and on some computations carried out with the aid of GAP [7].

Preliminaries
The following definitions of an NMDS code of length n and dimension k over a finite field F q are equivalent to that given in the Introduction; see [5].Definition 1.A linear [n, k] code over 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 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 C over F q can be regarded as a projective system (i.e. a distinguished point set) C in a projective space PG(k − 1, q); see [15] for more details.
it satisfies the following conditions: (i) every k − 1 points in C span a hyperplane of PG(k − 1, q); (ii) there exists a hyperplane of PG(k − 1, q) containing exactly k points of C; (iii) every k + 1 points of C generate the whole PG(k − 1, q).
Thus, in this setting, an NMDS [n, k] code over F q is an ( Given an integer ν ≥ 1 and a prime power q = p h , consider the set C ν of all the curves of degree ν contained in the projective plane PG(2, q) over a finite field F q .Since any curve C ∈ C ν is uniquely determined by m + 1 = ν+2 2 parameters in F q , that is, the coefficients of its equation and the curve is unchanged if these parameters are multiplied by a common factor, then C ν can be regarded as a projective space PG(m, q) with homogeneous coordinates (a 0 :a 1 : • • • :a m ).We may also denote a curve 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 E and C be two distinct cubic curves meeting in a set S consisting of 9 points (counted with multiplicities).If D ⊂ PG(2, q) is any cubic curve containing all but one point of S , then C ∩ D = S .

Lifting point sets
The space C 3 consisting of all the cubics in PG(2, q) has projective dimension 9, hence 10 independent cubic curves are required to generate it.Let E be a non-singular cubic curve of equation f (X, Y, Z) = 0 over F q .A suitable basis B for C 3 , containing E , can be written by using the following polynomials: 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 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 F q -rational points of a carefully chosen elliptic curve is not necessarily limited to q + 1.
More in detail, the points of the curve E are mapped onto a curve Γ of PG(9, q) with the same number n of F q -rational points as E .Also Γ is the complete intersection of V 3 with the hyperplane Σ ∼ = PG(8, q) of equation X 0 = 0. Since for every cubic curve C of equation g(X, Y, Z) = 0 in PG(2, q), the defining polynomial is a linear combination of the elements of B, that is, it turns out that ν 2 3 (C ) is the complete intersection of V 3 with the hyperplane Π ⊂ PG(9, q) of equation which is distinct from Σ. Thus, every cubic curve C : g(X, Y, Z) = 0 of PG(2, q) corresponds to a hyperplane of equation ( 2).Back to PG(2, q), the set (ν 2 3 ) −1 (Π ∩ V 3 ) corresponds to a unique cubic curve C distinct from E , and, clearly, (ν 2 3 ) −1 (Π ∩ Γ) corresponds to C ∩ E .
Theorem 3.1.Suppose that E has n ≥ 9 points.Then the point set Γ is an (n; 9, 7)-set in Σ = PG(8, q).Proof.To prove the theorem it suffices to consider the mutual position of cubic curves in PG(2, q).
(i) Take eight distinct points P 1 , . . ., P 8 ∈ Γ and consider the correspond- Suppose that there is a t-dimensional net with t ≥ 2, say F , consisting of cubics through Q 1 , . . ., Q 8 .Then, from Theorem 2.1 there is a ninth point Q 9 ∈ E such that the points Q 1 , . . ., Q 9 are in the support of F .This implies that every further point Q 10 ∈ E \ {Q 1 , . . ., Q 9 } yields a (t − 1)-dimensional net consisting of cubics through Q 1 , . . ., Q 9 which are distinct from E and have ten points in common with it, contradicting Bézout's theorem.Hence, F must be a pencil of cubic curves in PG(2, q) including E and passing through Q 1 , . . ., Q 8 .Back to PG(9, q), we observe that F corresponds to a pencil of hyperplanes of PG(9, q) which meet in a unique 7-dimensional subspace ∆ such that {P 1 , . . ., P 8 } ⊂ (Γ ∩ ∆), that is, P 1 , . . ., P 8 , generate the hyperplane ∆ of Σ.
(ii) From Theorem 2.1, there is a further point Q 9 ∈ PG(2, q) which belongs to the intersection of E and all the other cubics of the above pencil F .This proves that the previous subspace ∆ meets Γ in P 1 ,. . .,P 8 , (iii) Let Π be a hyperplane of PG(9, q) different from Σ. Put C = (ν 2 3 ) −1 (Π).From Bézout's theorem we know that |E ∩ C | ≤ 9, therefore any hyperplane of PG(9, q) has at most 9 points in common with Γ.Hence, Γ is a curve of order 9, therefore 10 points of Γ generate the whole Σ.

The claim follows.
Remark 1.The code associated to Γ can also be interpreted as an AG-code, see [15].Indeed, Theorem 3.1 is a consequence of [15,Theorem 4.4.19].However, our proof does not use the Riemmman-Roch Theorem.

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 E in the case when the base field is large enough.
By Definition 4, the algebraic curve Γ = ν 2 3 (E ) provides a complete NMDS code, that is a complete (n; 9, 7)-set of PG(8, q), if and only if for any Q ∈ Σ there exists at least one hyperplane Π of Σ with Q ∈ Π meeting Γ in 9 points.
We expect that for large q special points, if they exist at all, are very few; see Lemma 4.4.So we propose the following conjecture.
Conjecture 1. Suppose q ≥ 121 to be such that 2, 3 |q.Then there are no special points for Γ.
In order to verify Conjecture 1, we performed some computer searches for some values of q.For q ∈ {7, 11, 13} we executed a (non-trivial) exhaustive search.For q ≥ 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 V 3 are linearly independent; see [11].
For small values of q it is possible to perform an exhaustive search, adopting the following procedure: 3 (E ) be the embedding of E ; 2. for any set of 9 points of Γ, consider the matrix containing their components; let G be the list of such matrices having rank 8.In particular, each element of G corresponds to a hyperplane meeting Γ in 9 points.We call such hyperplanes good.

3.
For each matrix H ∈ 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 ′ = 0. 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 E of PG(2, q) admits an equation in canonical Weierstrass form with a, b ∈ F q such that −16(4a 3 + 27b 2 ) = 0; see [14].
Remark 2. Good hyperplanes correspond to linear systems of cubic curves cutting E in 9 points; by [12, Theorem 43], we see that the number of such hyperplanes is approximately 1 9! q 7 .We leave to a future work to determine exactly what sets of 9 distinct points of a given elliptic curve 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 E is 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 ∈ {7, 11, 13}.Then, the lifted (n; 9, 7)-set Γ in PG(8, q) is complete if and only if n = |E | ≥ 15.In particular, for q = 7 the lifted set Γ 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 Γ in Σ = PG(8, q) is complete and gives evidence supporting Conjecture 1.
As in the previous section, the projective plane P G(2, q) is assumed to be of order q odd and not divisible by 3. Furthermore we suppose q ≥ 121.Let j(E ) be the j-invariant of E , that is the six cross-ratios of the four tangents from a point of E to other points of E .We limit ourselves to the case j(E ) = 0, see [10,Theorem 11.15].
We will use the following result which is a direct consequence of [8, Lemma 3.2].Lemma 4.3.Let q ≥ 121 and consider an elliptic cubic E (F q ) with j(E ) = 0. Then there are at least 7 trisecant F q -rational lines through any given F qrational point.
3 (P ) with P ∈ E .Consider a reducible cubic curve C in PG(2, q), union of 3 lines ℓ, m, r with P ∈ ℓ \ {m ∪ r} and such that |(ℓ ∪ m ∪ r) ∩ E | = 9.Such a curve if |E | > 9 is guaranteed to exist by Lemma 4.3 and it corresponds to a hyperplane of PG(9, q) through Q meeting Γ in 9 distinct points.So Q is not special.Now consider a cubic curve C in PG(2, q) with equation of the form and a cubic curve C ′ with equation of type Via the Veronese embedding ν 2 3 , C corresponds to the hyperplane of equation αX 4 +βX 7 +γX 8 = 0, whereas C ′ corresponds to the hyperplane aX 1 +bX 3 + cX 4 = 0.
For any Q ∈ Σ \ Γ write P Q := [q 4 , q 7 , q 8 ] and P ′ Q := [q 1 , q 3 , q 4 ] ∈ PG(2, q).If P Q ∈ E , by Lemma 4.3 there are at least 7 lines through P Q meeting E in 3 distinct points; in particular there is at least one line of equation αX Consequently the cubic C : Y Z(αX + βY + γZ) = 0 corresponds to a hyperplane Π of PG(9, q) through Q, meeting Γ in 9 distinct points and we are done.
If P Q ∈ E but P ′ Q ∈ E , repeating the same argument starting from a cubic C ′ with equation (4), we see that Q is not special.
Thus, we suppose P Q , P ′ Q ∈ E and distinguish several cases: 1.If q 4 = 0, then the cubic C of equation XY Z = 0 corresponds to the hyperplane X 4 = 0 passing through Q with 9 intersections with Γ.
Let now q ≡ 1 mod 3 and ω be a root of T 2 + T + 1 = 0 Consider a non-singular plane cubic curve E over F q with canonical equation: where c = ∞, 1, ω, ω 2 .
If c = 1 + √ 3, then the elliptic curve E is harmonic that is, j(E ) = 0, see [10,Lemma 11.47].Using Remark 3 and the symmetry Y ↔ Z of the curve E it is possible to test for the completeness of ν 2 3 (E ).With the aid of GAP [7], 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.