Until the ’s, most Goppa codes were associated to curves. In 2001 S.H. Hansen 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  and further investigated by D. Joyner , J. Little and H. Schenck , D. Ruano  and I. Soprunov and J. Soprunova , 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  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 . A. Couvreur and I. Duursma  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  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 , 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  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  and Carvalho and Neumann’s works  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:
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  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.
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  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 , 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  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).
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.
The Picard group of the Hirzebruch surface is the free Abelian group
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
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 ( Theorem ). This description is given for instance by M. Reid .
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 ().
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:
The evaluation code is defined as the image of the evaluation map
Note that this code is Hamming equivalent to the Goppa code , as defined by Hansen . 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 .
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 ofin . 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 .
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 (, , , , ). It will be useful to handle the computation of the dimension and the minimum distance as a combinatorial problem.
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 and or .
Note that is a lattice polygone except if and does not divide .
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
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
The equivalence relation on and the bijection
endow with a equivalence relation, also denoted by , such that
Let two couples and be in and let us write
Then if and only if
It remains the case of and . We have
and divides and . Then it also divides . Evaluating at yields like previously of and only if . ∎
The conditions of Lemma 2.2.5 also can be written
Besides, the conditions involving are always satisfied for .
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 .
Let and . Let us set
Notice that is nothing but cut out by the set
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 .
Motivated by Example 2.2.9, we give a map that displays as a set of representatives of under the equivalence relation .
Let us set the map such that for every couple its image is defined as follows.
If or , then ,
Otherwise, we choose with .
If , then ,
If , then ,
Otherwise, we choose with .
The map induces a bijection from the quotient set to .
The set is a set of representatives of under the equivalence relation .
The set is a set of representatives of under the equivalence relation .
First notice that elements of are invariant under .
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.
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). ∎
The number of equivalence classes of under is equal to the cardinality of .
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