Algebraic Geometric codes on Hirzebruch surfaces

01/25/2018 ∙ by Jade Nardi, et al. ∙ Université de Toulouse 0

We define a linear code C_η(δ_T,δ_X) by evaluating polynomials of bidegree (δ_T,δ_X) in the Cox ring on F_q-rational points of the Hirzebruch surface of parameter η on the finite field F_q. We give explicit parameters of the code, notably using Gröbner bases. The minimum distance provides an upper bound of the number of F_q-rational points of a non-filling curve on a Hirzebruch surface. We also display some punctured codes having optimal parameters.

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Introduction

Until the ’s, most Goppa codes were associated to curves. In 2001 S.H. Hansen [8]estimated parameters of Goppa codes associated to normal projective varieties of dimension at least . As Hansen required very few assumptions on the varieties, the parameters he gave depended only on the Seshadri constant of the line bundle, which is hard to compute in practice. New classes of error correcting codes have thus been constructed, focusing on specific well-known families of varieties to better grasp the parameters. Among Goppa codes associated to a surface which have been studied so far, some toric and projective codes are based on Hirzebruch surfaces.

Toric codes, first introduced by J. P. Hansen [7] and further investigated by D. Joyner [9], J. Little and H. Schenck [12], D. Ruano [14] and I. Soprunov and J. Soprunova [15], are Goppa codes on toric varieties evaluating global sections of a line bundle at the -rational points of the torus. J. Little and H. Schenck [12] already computed the parameters of toric codes on Hirzeburch surfaces for some bidegrees and for large enough to make the evaluation map injective.

Projective codes evaluate homogeneous polynomials on the rational points of a variety embedded in a projective space. A first example of projective codes is the family of Reed-Muller projective codes on [10]. A. Couvreur and I. Duursma [2] studied codes on the biprojective space embedded in

. The authors took advantage of the product structure of the variety, yielding a description of the code as a tensor product of two well understood Reed-Muller codes on

. More recently C. Carvalho and V. G.L. Neumann [1] examined the case of rational surface scrolls as subvarieties of , which extends the result on , isomorphic to .

In this paper we establish the parameters of Goppa codes corresponding to complete linear systems on minimal Hirzebruch surfaces , a family of projective toric surfaces indexed by . This framework expands preceding works while taking advantage of both toric and projective features.

Regarding toric codes, we extend the evaluation map on the whole toric variety. This is analogous to the extension of affine Reed-Muller codes by projective ones introduced by G. Lachaud [10], since we also evaluate at "points at infinity". In other words toric codes on Hirzebruch surfaces can be obtained by puncturing the codes studied here at the points lying on the torus-invariant divisors, that have at least one zero coordinate. As in the Reed-Muller case, through the extension process, the length turns to grow about twice as much as the minimal distance, as proved in Section 6.

Respecting the projective codes cited above, it turns out that rational surface scrolls are the range of some projective embeddings of a Hirzeburch surface, for and for . However no embedding of the Hirzebruch surface into a projective space is required for our study and the Cox ring replaces the usual used in the projective context. Moreover, the embedded point of view forces to only evaluate polynomials of the Cox ring that are pullbacks of homogeneous polynomials of under this embedding. No such constraint appears using the Cox ring and polynomials of any bidegree can be examined.

Whereas coding theorists consider evaluation codes with an injective evaluation map, C. Carvalho and V. G.L. Neumann (loc. cit.) extensively studied codes associated to a non necessarily injective evaluation map. In the present work no assumption of injectivity is needed. In particular, the computation of the dimension of the code does not follow from Riemann-Roch theorem. For a given degree, this grants us a wider range of possible sizes for the alphabet, including the small ones.

Our study focuses on minimal Hirzeburch surfaces, putting aside , the blown-up of at a point. Although most techniques can be used to tackle this case, some key arguments fail, especially when estimating the minimal distance.

The linear code is defined as the evaluation code on -rational points of of the set of homogeneous polynomials of bidegree , defined in Section 1. The evaluation is naively not well-defined for a polynomial but a meaningful definition à la Lachaud [10] is given in Paragraph 1.2.

Here the parameters of the code are displayed as nice combinatoric quantities, from which quite intricate but explicit formulae can be deduced in Propositions 2.4.1 and 4.2.3. The rephrasing of the problem in combinatorial terms is already a key feature in Hansen’s [7] and Carvalho and Neumann’s works [1] that is readjusted here to fit a wider range of codes.

A natural way to handle the dimension of these codes is to calculate the number of classes under the equivalence relation on the set that identifies two polynomials if they have the same evaluation on every -rational point of the Hirzebruch surface. Our strategy is to first restrict the equivalence relation on the set of monomials of and a handy characterization for two monomials to be equivalent is given.

In most cases comprehending the equivalence relation over monomials is enough to compute the dimension. We have to distinguish a particular case:

Theorem A.

The dimension of the code satisfies

where is equal to if the couple satisfies (Introduction) and otherwise.

This quantity depends on the parameter , the bidegree and the size of the finite field.

As for the dimension, the first step to determine the minimum distance is to bound it by below with a quantity that only depends on monomials. Again the strategy is similar to Carvalho and Neumann’s one [1] but, even though they mentioned Gröbner bases, they did not fully benefit from the potential of the tools provided by Gröbner bases theory. Indeed linear codes naturally involve linear algebra but the problem can be considered from a commutative algebra perspective. On this purpose, we consider the homogeneous vanishing ideal of the subvariety constituted by the -rational points. A good understanding of a Gröbner basis of , through Section 3, shortens the proof of the following theorem.

Theorem B.

Let us fix such that . The minimum distance satisfies

where is defined in Notation 4.1.1. It is an equality for and .

The cardinality of depends on the parameter , the bidegree and the size of the finite field.

The pullback of homogeneous polynomials of degree on studied by C. Carvalho and V. G.L. Neumann are polynomials of bidegree on . C. Carvalho and V. G.L. Neumann gave a lower bound of the minimum distance that we prove to be reached since it matches the parameters we establish here. The parameters also coincide with the one given by A. Couvreur and I. Duursma [2] in the case of the biprojective space , isomorphic to Hirzebruch surface .

It is worth pointing out that the codes with negative have never been studied until now. Although this case is intricate when the parameter divides and the situation (Introduction) occurs, it brings the ideal to light as an example of a non binomial ideal on the toric variety .

The last section highlights an interesting feature of these codes which leads to a good puncturing. It results codes of length but with identical dimension and minimum distance.

We emphasize that the lower bound of the minimum distance in this paper does not result from upper bound of the number of rational point of embedded curves but from purely algebraic and combinatoric considerations. This approach, already highlighted by Couvreur and Duursma [2], stands out from the general idea that one would estimate the parameters of an evaluation code on a variety though the knowledge of features of , like some cohomology groups for the dimension or the number of rational points of subvarieties of for the minimum distance. It also offers the great perspective of solving geometric problems thanks to coding theory results. Moreover, the non injectivity of the evaluation map means that there exists a filling curve, i.e. a curve that contains every -rational point of . From a number theoretical point of view, the minimum distance provides an upper bound of the number of -rational points of a non filling curve, regardless of its geometry and its smoothness, even if there exist some filling curves.

1 Defining evaluation codes on Hirzebruch surfaces

1.1 Hirzebruch surfaces

We gather here some results about Hirzebruch surfaces over a field , given in [4] for instance.

Let be a non negative integer. The Hirzebruch surface can be considered from different points of view.

On one hand, the Hirzebruch surface is the toric variety corresponding to the fan (see Figure 1).

Figure 1: Fan

The fan being a refining of the one of , it yields a ruling of fiber and section . The torus-invariant divisors , , and corresponding to the rays spanned respectively by , , , generate the Picard group of , described in the following proposition.

Proposition 1.1.1.

The Picard group of the Hirzebruch surface is the free Abelian group

where

(1)

We have the following intersection matrix.

As a simplicial toric variety, the surface considered over carries a Cox ring . Each monomial of is associated to a torus-invariant divisor

(2)

The degree of the monomial is defined as the Picard class of the divisor . The couple of coordinates of in the basis is called the bidegree of and denoted by . By (1) and (2),

(3)

It is convenient to set

This gives the -grading on

where is the -module of homogeneous polynomials of bidegree . Note that the -module is non zero if and only if and .

On the other hand, the Hirzebruch surface can be displayed as a geometric quotient of an affine variety under the action of an algebraic group ([4] Theorem ). This description is given for instance by M. Reid [13].

Let us define an action of the product of multiplicative groups over : write for the first coordinates on , on the second coordinates on and for elements of . The action is given as follows:

Then the Hirzebruch surface is isomorphic to the geometric quotient

This description enables us to describe a point of by its homogeneous coordinates .


In this paper, we focus only on minimal Hirzebruch surfaces. A surface is minimal if it contains no curve. We recall the following well-known result about minimal Hirzebruch surface.

Theorem 1.1.2 ([11]).

The Hirzeburch surface is minimal if and only if .

1.2 Evaluation map

We consider now the case , being a power of a prime integer.

From the ruling , the number of rational points of the Hirzebruch surface is

Let such that . Given a polynomial and a point of , the evaluation of at is defined by , where is the only tuple that belongs to the orbit of under the action of and has one of these forms:

  • with ,

  • with ,

  • with ,

  • .

The evaluation code is defined as the image of the evaluation map

(4)

Note that this code is Hamming equivalent to the Goppa code , as defined by Hansen [8]. The weight of a codeword is the number of non-zero coordinates. The minimum weight among all the non-zero codewords is called the minimum distance of the code and is denoted by .

2 Dimension of the evaluation code on the Hirzebruch surface

Let us consider and such that .

Notation 2.0.1.

The kernel of the map is denoted by .

From the classical isomorphism

the dimension of the evaluation code

equals the dimension of any complementary vector space of

in . This is tantamount to compute the range of a well-chosen projection map on along .

2.1 Focus on monomials

The aim of this section is to display a projection map, denoted by , that would have the good property of mapping a monomial onto a monomial. The existence of such a projection is not true in full generality: given a vector subspace of a vector space and a basis of , it is not always possible to find a basis of composed of difference of elements of and a complementary space of which basis is a subset of . This will be possible here except if (Introduction) holds.

With this goal in mind, our strategy is to focus first on monomials of . Let us define the following equivalence relation on the set of monomials of .

Definition 2.1.1.

Let us define a binary relation on the set of monomials of . Let . We note if they have the same evaluation at every -rational point of , i.e.

This section is intended to prove that, even if this equivalence relation can be defined over all , the number of equivalence classes when considering all polynomials is the same as when regarding only monomials, unless (Introduction) holds. This section thus goals to prove Theorem A, stated in the introduction.

2.2 Combinatorial point of view of the equivalence relation on monomials

Throughout this article, the set are pictured as a polygon is of coordinates . This point of view, inherited directly from the toric structure, is common in the study of toric codes ([7], [9], [14], [12], [15]). It will be useful to handle the computation of the dimension and the minimum distance as a combinatorial problem.

Definition 2.2.1.

Let . Let us define the polygon

associated to the divisor and

Being intersection of with half planes, it is easily seen that is the set of lattice points of the polygon , which vertices are

  • if ,

  • if and or .

Note that is a lattice polygone except if and does not divide .

Notation 2.2.2.

Let us set

the -coordinate of the right-most vertices of the polygon .

Let us highlight that is not necessary an integer if . Thus it does not always appear as the first coordinate of an element of . It is the case if and only if . If so, the only element of such that is its first coordinate is .

We thus observe that

(5)

(a)
e.g.

(b) ,
e.g. in

(c) ,
e.g. in
Figure 2: Different shapes of the polygon
Example 2.2.3.

Figure 2 gives the three examples of possible shapes of the polygon . The first one is the case , and the last two ones correspond to and depend on the sign of , which determines the shape of . All proofs of explicit formulae in Propositions 2.4.1 and 4.2.3 distinguish these cases.

Thanks to (3), a monomial of is entirely determined by the couple . Then each element of corresponds to a unique monomial. More accurately, for any couple , we define the monomial

(6)
Definition 2.2.4.

The equivalence relation on and the bijection

(7)

endow with a equivalence relation, also denoted by , such that

Proposition 2.2.5.

Let two couples and be in and let us write

Then if and only if

(C1)
(C2)
(C3)
(C4)
Proof.

The conditions (C1), (C2), (C3) and (C4) clearly imply that , hence . To prove the converse, assume that and write

Let . Then , which means . But this equality is true for any element of if and only if . This is equivalent to or and , which proves (C2) and (C4) for .

Repeating this argument evaluating at for every gives and if and only if , i.e. (C1) and (C3) for .

Moreover, we have , which means that if and only if . Evaluating at gives . Then if and only if . This proves (C1) and (C3) for .

It remains the case of and . We have

and divides and . Then it also divides . Evaluating at yields like previously of and only if . ∎

Remark 2.2.6.

The conditions of Lemma 2.2.5 also can be written

(8)
(9)

Besides, the conditions involving are always satisfied for .

Observation 2.2.7.

The conditions (C3) and (C4) mean that a point of lying on an edge of can be equivalent only with a point lying on the same edge. Therefore the equivalence class of a vertex of is a singleton.

To prove that the number of equivalence classes equals the dimension of the code as stated in Theorem A (unless (Introduction) holds), we goal to choose a set of representatives of the equivalence classes of under the relation , which naturally gives a set of representatives for under the binary relation .

Notation 2.2.8.

Let and . Let us set

Notice that is nothing but cut out by the set

Example 2.2.9.

Let us set and . Let us sort the monomials of , grouping the ones with the same image under , using Proposition 2.2.5.

Figure 3 represents the set . Note that for each couple , there is exactly one of these groups that contains the monomial .

Figure 3: Dots in correspond to elements of .

Motivated by Example 2.2.9, we give a map that displays as a set of representatives of under the equivalence relation .

Definition 2.2.10.

Let us set the map such that for every couple its image is defined as follows.

  • If or , then ,

  • Otherwise, we choose with .

and

  • If , then ,

  • If , then ,

  • Otherwise, we choose with .

Proposition 2.2.11.
  1. The map induces a bijection from the quotient set to .

  2. The set is a set of representatives of under the equivalence relation .

  3. The set is a set of representatives of under the equivalence relation .

Proof.

First notice that elements of are invariant under .

The inclusion is clear by definitions of (Not. 2.2.8) and (Def. 2.2.10). The equality follows from the invariance of .

Last, we prove that for every couple . Take a couple and denote by its image under . We have to prove that and satisfy all the conditions of Proposition 2.2.5.

By definition of , it is clear that conditions (C1), (C2), (C3), as well as the the forward implication of (C4), are true. It remains to prove that for .

Let us prove only the case . So assume that . Then or . However,

This is only possible when and then . By condition (C3), this implies that and then . This proves the first item.

The second assertion is a straightforward consequence of the first one.

Finally the third assertion yields from the definition of the equivalence relation on via the bijection (7). ∎

Corollary 2.2.12.

The number of equivalence classes of under is equal to the cardinality of .

Proof.

Its results from Definition 2.2.4 and Proposition 2.2.11. ∎

2.3 Proof of Theorem A

The main idea of the proof is to define an endomorphism on the basis of monomials by conjugation of