Minimum weight codewords in dual Algebraic-Geometric codes from the Giulietti-Korchmáros curve

In this paper we investigate the number of minimum weight codewords of some dual Algebraic-Geometric codes associated with the Giulietti-Korchmáros maximal curve, by computing the maximal number of intersections between the Giulietti-Korchmáros curve and lines, plane conics and plane cubics.

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1 Introduction

Algebraic-Geometric codes (AG codes for short) are an important class of error correcting codes; see [10, 11, 19].

Let be an algebraic curve defined over the finite field of order . The parameters of the AG codes associated with strictly depend on some characteristics of the underlying curve . In general, curves with many -rational places with respect to their genus give rise to AG codes with good parameters. For this reason maximal curves, that is curves attaining the Hasse-Weil upper bound, have been widely investigated in the literature: for example the Hermitian curve and its quotients, the Suzuki curve, and the Klein quartic; see [12, 14, 15, 18, 20, 22, 21, 23, 3]. More recently, AG codes were obtained from the Giulietti-Korchmáros curve [9] (GK curve for short), which is the first example of maximal curve shown not to be covered by the Hermitian curve; see [7, 4, 2].

In general, to know the weight distribution of a particular code is a hard problem. Even the problem of computing codewords of minimum weight can be a difficult task apart from specific cases. In [13], following the approach of [16, 1], the authors compute the number of minimum weight codewords of some dual AG codes from the Hermitian curve by providing an algebraic and geometric description for codewords of a given weight belonging to any fixed affine-variety code.

In this work we deal with AG codes arising from the GK maximal curve. The link between the minimum distance of such codes and the underlying curve is given by a result of [5]; see Theorem 2.1.

In Section 2 we introduce basic notions and preliminary results concerning the GK curve, AG codes, and affine variety codes. In Section 3 we compute the maximal intersections between the GK curve and lines, plane conics, and plane cubics. Such information is used in Section 4 to compute the number of minimum weight codewords of some dual codes from the GK curve.

2 Preliminary Results

2.1 The Giulietti-Korchmáros curve

Let , prime and . Denote by the three dimensional projective space over the field with element. The Giulietti-Korchmáros curve (GK curve for short) is a non-singular curve in , introduced in [9], defined by the affine equations

 {Zℓ2−ℓ+1=Yℓ2−YYℓ+1=Xℓ+X. (1)

This curve has genus , -rationals points and a unique point at the infinity . The set of the -rational points splits into two orbits under the action of : the first one is composed by the set of the -rational points of , coinciding with the intersection between and the plane ; the second one is formed by all the points in . The GK curve is -maximal, that is, it attains the Hasse-Weil bound ; see [19, Theorem 5.2.3]. Moreover, for , is not covered by the Hermitian curve (see [9]): this is the first example in the literature of a family of maximal curves with this feature.

An algebraic curve contained in a projective space of dimension is said to be a complete intersection if the ideal associated with is generated by exactly polynomials. The curve is an example of a complete intersection curve in .

Consider now the function field associated with (see [19] for the connection between function fields and curves) and let be its coordinate functions, which satisfy and .

A divisor on is a formal sum , where , , if . The divisor is -rational if it coincides with its image under the Frobenius map over . For a function , indicates the divisor of .

Concerning the functions it is easily proved that

• ,

• ,

• ,

where denotes the affine point and .

2.2 Algebraic-Geometric codes

In this section we introduce some basics notions on AG codes. For a detailed introduction we refer to [19].

Let be a projective curve over the finite field , consider the rational function field and the set given by the -rational places of . Given a -rational divisor the Riemann-Roch space associated to on

is the vector space

over is defined as

 L(D)={f∈Fq(X) | (f)+D≥0}∪{0}.

It is known that this is a finite dimensional -vector space and the exact dimension can be computed using Riemann-Roch theorem.

Consider now the divisor where all the s have weight one. Let be another -rational divisor such that . Consider the evaluation map

 eD:L(G)→Fnqf⟼eD(f)=(f(P1,…,f(Pn)).

This map is -linear and it is injective if .

The AG-code associated with the divisors and is then defined as . It is an code, where and is the so called designed minimum distance of the code.

The differential code is defined as

 CΩ(D,G):={(resP1(ω),…,resPn(ω))|ω∈Ω(G−D)},

where . The differential code is an code, where and denotes the dual designed minimum distance.

The aim of this paper is to find the minimum distance of some dual Algebraic-Geometric codes. To achieve this goal we will use a number of times the following result which is a byproduct of [5, Theorem 3.5].

Theorem 2.1.

Let be a non singular curve which is complete intersection in a projective space of dimension , the divisor , an integer, and the minimum distance of the code . Then

1. if and only if points of are collinear in ;

2. if and only if no points of are collinear and there exist points of lying on a plane conic (possibly reducible);

3. if and only if no points of are collinear, no points lie on a plane conic, and there exist points of coplanar and belonging to the intersection of a cubic curve and a curve of degree having no common irreducible component;

4. if and only if no sub-family of the points of satisfies one of the three above configurations.

2.3 Affine variety codes

We introduce now affine variety codes, see [8] for further information.

Let and consider an ideal of , . The ideal is zero-dimensional and radical; see [17]. Let be the variety of and .

Definition 2.2.

An affine variety code is the image of , a -vector subspace of of dimension , given by the isomorphism of -vector spaces:

 ϕ:R⟶Fnqf⟼(f(P1),…,f(Pn)).

Let be generated by , then the matrix

 H:=⎛⎜ ⎜ ⎜ ⎜ ⎜⎝b1(P1)b1(P2)…b1(Pn)b2(P1)b2(P2)…b2(Pn)⋮⋮⋮⋮br(P1)br(P2)…br(Pn)⎞⎟ ⎟ ⎟ ⎟ ⎟⎠

is a generator matrix for and a parity-check matrix for . It is clear that there is a strong connection between affine variety codes and Algebraic-Geometric codes and that, depending on the choice of , they can coincide.

Since we are interested in computing the number of minimum weight codewords of particular AG codes, next proposition will give us a useful criterion.

Proposition 2.3 (Marcolla, Pellegrini, Sala, [13]).

Let . Let be such that . Let be a subspace of of dimension generated by . Let be the ideal in generated by

 ∑wi=1zibj(xi,1,…,xi,t)%forj=1,…,r,gh(xi,1,…,xi,t)for i=1,…,wand h=1,…,s,zq−1i−1for i=1,…,w,∏1≤l≤t((xj,l−xi,l)q−1−1)for 1≤j

Then any solution of corresponds to a codeword of of weight . Also, the number of codewords of weight is

 Aw(C(I,L)⊥)=|V(Jw)|w!,

where is the number of distinct solutions of .

3 Intersection between the GK curve and lines or conics

In this section we study the possible intersections between a line or a plane conic and the curve as in (1). In particular, we are interested in the maximum possible size of their intersections.

Proposition 3.1.

Let be a line. Then

 |r∩GK|≤ℓ2−ℓ+1.

Also, any -secant is parallel to the -axis and all the common points are not -rational.

Proof.

As already mentioned, the -rational points of are divided into two orbits and .

Suppose that contains at least an -rational point . Without loss of generality we can assume that . Let . This implies . An -rational point on the line through and has coordinates

 (λx1+λ,λy1+λ,λz1+λ),

for some . If such a point belongs to then

 (λy1+λ)ℓ+1=(λx1+λ)ℓ+λx1+λ,

that is

 λℓ+1yℓ+1=λℓxℓ(1+λ)+λx(1+λ)ℓ.

The condition yields Therefore it is easily seen that the number of the intersections between the line and the GK curve is exactly .

By direct checking, the same happens for the line through and .

Suppose now that contains no points of . Let two points of . An -rational point of is

 P=(x1+λx21+λ,y1+λy21+λ,z1+λz21+λ)

for some . If then by the second equation in (1)

 (y1+λy21+λ)ℓ+1=(x1+λx21+λ)ℓ+x1+λx21+λ.

Recalling that and , we obtain

 λℓ(x1+xℓ2−y1yℓ2)+λ(xℓ1+x2−yℓ1y2)=0.

If or then . On the other hand, if then

 (x1+xℓ1)+(x2+xℓ2)−y1yℓ2−yℓ1y2=0,yℓ+11+yℓ+12−y1yℓ2−yℓ1y2=(y1−y2)ℓ+1=0,

that is . Finally, from , we get . This means that if then has equation , , with . Clearly otherwise and belong to . By direct checking, the line has exactly points in common with the curve .

Proposition 3.2.

The total number of -secants of the is .

Proof.

Recall that . Also, each point in lies on exactly one -secant such that . Therefore the number of such lines is

 (ℓ8−ℓ6+ℓ5−ℓ3)(ℓ2−ℓ+1)=(ℓ+1)(ℓ5−ℓ3).

Proposition 3.3.

Let be a plane conic in . Then the size is at most

 {2(ℓ2−ℓ+1) if C reducible,2(ℓ+1) if C absolutely irreducible.
Proof.

Let be contained in the plane defined by . Suppose that is absolutely irreducible.

Suppose . The points in satisfy

 ⎧⎪ ⎪ ⎪ ⎪⎨⎪ ⎪ ⎪ ⎪⎩zℓ2−ℓ+1=yℓ2−yyℓ+1=xℓ+xax2+by2+cxy+dx+ey+f=0G(x,y,z)=0,

where . By Bézout’s Theorem the number of pairs satisfying and is at most . Clearly, for each such pair there exists a unique satisfying . Therefore .

Suppose now and . The points in satisfy

 ⎧⎪ ⎪ ⎪ ⎪⎨⎪ ⎪ ⎪ ⎪⎩zℓ2−ℓ+1=yℓ2−yyℓ+1=xℓ+xax2+bz2+cxz+dx+ez+f=0G(x,y)=0,

where . As above, there are at most pairs such that and . Clearly, for each such pair there exist at most values such that , since the is absolutely irreducible. Therefore .

The case and is similar and omitted.

If the conic splits into two lines, then by Proposition 3.1 it is clear that is at most . Note that if the two lines are -secants then their common point is . ∎

The previous result can be generalized to a plane curve of degree .

Proposition 3.4.

Let be a curve of degree in . Then the size is at most

 {α(ℓ2−ℓ+1) if X reducible,α(ℓ+1) if X absolutely irreducible.

We conclude this section with the following proposition.

Proposition 3.5.

There exist coplanar points contained in lying on the intersection between a cubic curve and a curve of degree .

Proof.

Let . Consider three lines of equations , , , with . Such three lines are coplanar and -secants; see also Proposition 3.1.

Let be the plane cubic consisting of the union of , , and . Clearly, . To conclude the proof we have to show that these points lie on a plane curve of degree .

It is enough to observe that the points in are

 (xi,¯¯¯y,zj),i=1,2,3, j=1,…,ℓ2−ℓ+1,

with . Such points are contained in the lines of equations

 sj:={Y=¯¯¯yZ=zj.

Therefore is contained also in .

4 Minimum distance and number of minimum weight codewords of one point codes on the GK curve

We first determine the minimum distance of the one point AG code , where and .

Proposition 4.1.

The minimum distance of is

1. when ;

2. when ;

3. when ;

4. when .

Proof.

We apply Theorem 2.1.

1. By Proposition 3.1 there exist collinear points in and therefore the minimum distance is .

2. If then points of cannot be collinear. Since there exist points contained in a reducible plane conic (see Proposition 3.3) the minimum distance is exactly .

3. If then no line contains points and no plane conic contains points of . By Proposition 3.5 there exist plane cubics with points which are also contained in a curve of degree having no common components with the cubic. Therefore the minimum distance is .

4. If , none of the previous cases applies and therefore the minimum distance is at least .

Corollary 4.2.

Let be the designed Goppa minimum distance of . The minimum distance of is

 d⎧⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎨⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎩m+2,m≤ℓ2−ℓ−1;2m+2,m=ℓ2−ℓ;3m,m=ℓ2−ℓ+1;≥3m+1,ℓ2−ℓ+1
Proof.

It is enough to observe that is larger than the designed minimum distance only when

 3m+1≥m(ℓ3+1)−ℓ5+2ℓ3−ℓ2+2⟺m≤ℓ2−2+3ℓ2−5ℓ3−2⟺m≤ℓ2−1.

4.1 Number of minimum weight codewords

In this subsection we determine the number of minimum weight codewords in in the case .

Recall that for the code the designed Goppa minimum distance is

 d∗=deg(mP∞)−2g(GK)+2=m(ℓ3+1)−ℓ5+2ℓ3−ℓ2+2.

Consider the ideal and let . Also, let

 Bℓ,m={XiYjZk+I | i∈[0,…,ℓ−1], j∈[0,…,ℓ2−ℓ], k∈[0,…,m]}

and . By [7], is a basis for the Riemann-Roch space .

To count the exact number of the minimum weight codewords of we use Proposition 2.3. Let . Using the same notations, we consider the ideal given by

A point in is a -tuple

 (¯x1,…,¯xw,¯y1,…,¯yw,¯z1,…,¯zw,¯u1,…,¯uw)∈F4wℓ6

which corresponds to a set of points , , in .

Theorem 4.3.

Let . The number of minimum weight codewords in is

Proof.

By Proposition 2.3, we have to count the number -uples

 (¯x1,…,¯xd,¯y1,…,¯yd,¯z1,…,¯zd,¯u1,…,¯ud)∈F4dℓ6

which differ in the first coordinates, and such that , , and

 ⎧⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎨⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎩¯u1+⋯+¯ud=0¯x1¯u1+⋯+¯xd¯ud=0¯y1¯u1+⋯+¯yd¯ud=0¯z1¯u1+⋯+¯zd¯ud=0⋮¯xℓ−11¯yℓ2−ℓ1zd−21¯u1+⋯+xℓ−1d¯yℓ2−ℓdzd−2d¯ud=0. (2)

To each tuple we can associate points , , in . Suppose that the number of different values is . Without loss of generality let be pairwise distinct.

Suppose . Let , for . We may suppose and let . Note that since then . System (2) contains the equations

 ¯yr1¯u1+⋯+¯yrd¯ud=0,¯x1¯yr1¯u1+⋯+¯xd¯yr1¯ud=0,⋮¯xℓ−11¯yr1¯u1+⋯+¯xℓ−1d¯yr1¯ud=0,

for . Let us define for , and , , . The above set of equations can be written as

 ⎧⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎨⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎩U1+⋯+Uα=0¯y1U1+⋯+¯yαUα=0¯y21U1+⋯+¯y2αUα=0⋮