Two-point AG codes from the Beelen-Montanucci maximal curve

In this paper we investigate two-point algebraic-geometry codes (AG codes) coming from the Beelen-Montanucci (BM) maximal curve. We study properties of certain two-point Weierstrass semigroups of the curve and use them for determining a lower bound on the minimum distance of such codes. AG codes with better parameters with respect to comparable two-point codes from the Garcia-G\"uneri-Stichtenoth (GGS) curve are discovered.


Introduction
Algebraic geometry codes, or simply AG codes, are a family of error-correcting codes introduced by Goppa in the '80s (see [10], [11]) and constructed using algebraic curves defined over a finite field. AG codes provide good examples of error-correcting codes when compared to other linear codes such as Reed-Solomon codes (RS codes). The basic parameters of an AG code C are its length n C , its dimension k C and its minimum distance d C . The minimum distance d C describes the error-correcting capability of the code, thus it is desirable for applications to construct codes with large minimum distance. A general lower bound for the minimum distance of an AG code is given by the well-known Goppa bound; a consequence of this bound is that for a code whose underlying algebraic curve has genus g, the inequality d C ≥ n C − k C + 1 − g holds, hence the minimum distance can be designed.
Let F q be the finite field with q elements and X be an algebraic curve defined over F q and of genus g. X is said to be maximal if it attains the Hasse-Weil bound |X (F q )| ≤ q + 1 + 2g √ q, where |X (F q )| is the number of F q -rational points of X . In other words, a maximal curve has the largest number of rational points with respect to its genus. For this reason maximal curves are suitable candidates for the construction of AG codes with good parameters.
An important example of maximal curve over a finite field is the Hermitian curve H q : y q+1 = x q+1 − 1, often defined by the equivalent affine equation y q+1 = x q + x, which is maximal over F q 2 . The Hermitian curve has been intensively studied and, together with the Suzuki and the Ree curves, it forms a family of maximal curves over suitable finite fields called the Deligne-Lusztig curves.
Another important example of maximal curve is the GK curve, constructed by Giulietti and Korchmáros in [9]. The GK curve is defined over F q 6 by the affine equations GK : y q+1 = x q + x, z q 2 −q+1 = y x q 2 −x x q +x . A generalization of the GK curve to an infinite family of F q 2n -maximal curves GGS n for n ≥ 3 odd has been given by Garcia, Güneri and Stichtenoth in [7]. The GGS n curves are defined by affine equations GGS n : where m := (q n + 1)/(q + 1). The GK curve is a special case of the GGS n curve for n = 3.
A different generalization of the GK curve was introduced by Beelen and Montanucci in [3]. For a prime power q and an odd integer n ≥ 3, the Beelen-Montanucci curve BM n is defined over F q 2n by the affine equations where m := (q n + 1)/(q + 1). The curve BM n is maximal over the field F q 2n and for n = 3 it is isomorphic to the GGS n curve and, equivalently, to the GK curve. Further, BM n is isomorphic to GGS n if and only if n = 3, as shown in [3]. For dual codes of two-point AG codes, methods that give a lower bound on the minimum distance that is possibly better than the Goppa bound were studied by Matthews in [13] and Beelen in [2]; both involve a generalization of the Weierstrass semigroup at one point to pairs of points. Over the years, these techniques have been applied to specific maximal curves: for example this is the case of the Hermitian curve in [12], [16] and [6], the Suzuki curve in [14], the GK curve in [5] and the GGS n curves in [1].
The aim of this paper is to study duals of two-point AG codes coming from the BM n curve. Our approach is similar to the one developed in [1], in which the GGS n curves are considered. In particular, the order bound introduced in [2] will be used to compute a lower bound on the minimum distance that improves the Goppa bound. In Section 2, general results on AG codes will be presented, with a particular focus on two-point AG codes. Section 3 is dedicated to the study of a certain two-point Weierstrass semigroup, as defined in [4], on the BM n curve. The fourth and last Section is devoted to the computation of the order bound for duals of two-point AG codes coming from the BM n curve and includes results for specific values of q and n.

Preliminary results
We recall some notations and results for two-point AG codes. A more general exposition of AG codes can be found in [17,Chapter 2]. Throughout this section, let q be any prime power. For an algebraic curve X defined over the finite field F q of genus g(X ), denote with F q (X ) the field of functions on X . For a fixed positive integer n, let P 1 , . . . , P n be rational points of X and let D be the divisor D := P 1 + · · · + P n . Further, let G be another divisor whose support is disjoint from the support of D. The Riemann-Roch space associated to G is the F q -vector space The AG code C L (D, G) is defined as is a linear subspace of F n q of dimension dim(C L (D, G)) = dim(L(G))− dim(L(G − D)) and its minimum distance d satisfies the bound d ≥ n − deg(G). Further, we define C L (D, G) ⊥ as the dual of C L (D, G); C L (D, G) ⊥ is a linear subspace of F n q of dimension n − dim(C L (D, G)) and its minimum distance d satisfies the bound d ≥ deg(G) − 2g(X ) + 2.
The lower bound d ≥ deg(G) − 2g(X ) + 2 is generally not tight and can be improved. To this aim, an approach similar to the one used in [2] can be used. We write G = F 1 + F 2 for some divisors F 1 and F 2 of X and we define for a rational point R of X : H(R; G) is called the set of G-non-gaps at R, studied for example in [8].
Observe that H(R; G + R) = H(R; G). Also, observe that H(R; 0) is the Weierstrass semigroup H(R) at R.
A convenient choice for the divisors F 1 and F 2 that we will keep for the rest of the paper is F 1 = 0 and F 2 = G. With these definitions in place, we recall now the generalized order bound introduced in [2, Section 2], to which we refer for a detailed discussion. Definition 6]). Let D be a divisor that is a sum of n distinct rational points of X and G a divisor on X such that all the points in its support are rational. Further, suppose that the support of G is disjoint from the support of D. For any infinite sequence S = R 1 , R 2 , . . . of points of X \ supp(D) define where the minimum is taken over all where the maximum is taken over all infinite sequences S of points having entries in X \ supp(D).
From [2,Theorem 7], the following proposition holds. By virtue of Proposition 2.2 we will refer to the quantity d(G) in Definition 2.1 as the generalized order bound for the minimum distance of C L (D, G) ⊥ or, simply, the order bound.
Since d S (G) does not depend on the chosen sequence S, the conclusion follows. Lemma 2.3 shows that the order bound d(G) coincides with the Goppa bound if G has degree larger than or equal to 4g − 1. At the same time, the order bound cannot be worse than the Goppa bound for deg(G) < 4g − 1.
Remark 2.4. Though the order bound d(G) can be obtained theoretically by considering all possible sequences of points that do not occur in the support of D, this is not feasible in practice unless we restrict the set of possible sequences to a finite set. A first step into this direction is to observe that the computation of d(G) using Definition 2.1 is only needed when deg(G) < 4g−1 (see Lemma 2.3) and that, in this case, only the first 4g − 1 − deg(G) entries from every sequence S are relevant to define d(G). However, some additional condition must be imposed: for example, at the cost of obtaining a possibly worse bound, one restricts the choice of the points that can occur in a sequence S to a finite set of points P, chosen beforehand. For practical convenience, the set P can be chosen as the set of rational points of X that are not in the support of D.
Throughout the rest of the paper we will apply the restriction suggested in Remark 2.4 when needed for practical purpose. With slight abuse of notation we will continue denoting the bound with d(G) and we will refer to it as the order bound. Note that this choice does not affect the statements of Proposition 2.2 and Lemma 2.3.
In Section 4 we will use Proposition 2.2 to obtain a lower bound on the minimum distance of duals of two-point AG codes. A two-point AG code is an AG code C L (D, G) such that the support of the divisor G consists of two distinct points Q and P only, namely G = aQ + bP for some a, b ∈ Z.
Remark 2.5. The dual of the two-point code C L (D, G) is not necessarily a two-point code; in fact, it is known that C L (D, G) ⊥ = C L (D, H) for some divisor H whose support is disjoint from the support of D, but the support of H might consist of more than two points. See [17, Proposition 2.2.10] for more details.
For a two-point code C L (D, G), we typically choose Q and P to be rational points of X and D to be the sum of all rational points of X different from Q and P . In such case, to the aim of computing the order bound d(G) on the minimum distance of the code C L (D, G) ⊥ , one can consider sequences We conclude this section with an exposition of a techinque, also used in [1], for conveniently computing the sets H(Q; bP ) and H(P ; aQ). Denote with R(Q, P ) the ring of functions in F q (X ) that are regular outside Q and P , namely The two-point Weierstrass semigroup H(Q, P ) can be defined as the set This generalization of the classical Weierstrass semigroup at one point has been studied for example in [4]. We define the period of the two-point Weierstrass semigroup H(Q, P ) as π := min{k ∈ N \ {0} | k(Q − P ) is a principal divisor} and we define the map Some of the properties of the map τ Q,P are summarized in the following proposition. See [4, Proposition 14, Proposition 17] for details.
Proposition 2.6. Let π be the period of the two-point Weierstrass semigroup H(Q, P ) and g = g(X ) be the genus of X . Then: For computational purposes, it is convenient to provide a method to describe the map τ −1 Q,P = τ P,Q . With the following proposition we show that τ −1 Q,P (j) can be computed efficiently for all j ∈ Z. Proposition 2.7. Let π be the period of the two-point Weierstrass semigroup H(Q, P ). Let j ∈ Z and i := τ −1 Q,P (j).
Finally, we recall two useful results that rely on the knowledge of the function τ Q,P only. The first one allows the determination of the dimension of the Riemann-Roch space of a two-point divisor G = aQ + bP , a, b ∈ N. The second one provides an explicit expression for the set of G-non-gaps at Q and the set of G-non-gaps at P . These results are crucial for the computation of the order bound.

The Weierstrass semigroup at a certain pair of points on the BM n curve
Let q be a prime power, n ≥ 3 be an odd integer and m := (q n + 1)/(q + 1). We devote this section to the study of a particular two-point Weierstrass semigroup, which will be specified later on, of the curve BM n defined in (2). Firstly, we summarize some of the main properties of the curve BM n . Further details can be found in [3].
Proposition 3.1. Let BM n be the Beelen-Montanucci curve defined in (2) and H q the Hermitian curve defined in (1).
• BM n is maximal over the field F q 2n .
• The q 3 + 1 F q 2 -rational points of H q are totally ramified in the cover BM n → H q .
• The full automorphism group Aut(BM n ) of BM n is isomorphic to SL(2, q) C q n +1 , where C q n +1 is the cyclic group with q n + 1 elements, and acts on the set of F q 2 -rational points of BM n with two orbits where P 1 , . . . , P q+1 are the q + 1 F q 2 -rational points of BM n lying over the q + 1 points at infinity of H q and Q 1 , . . . , Q q 3 −q are the q 3 − q F q 2rational points of BM n lying over the remaining F q 2 -rational points of H q .
Note that the points in O 1 can be parametrized in homogeneous coordinates (x : y : Since the Weierstrass semigroup at any point is invariant under the action of the automorphism group Aut(BM n ) on that point (see for example [17,Lemma 3.5.2]), points in the same orbit have the same Weierstrass semigroup. In the following, we will choose We recall some functions in F q 2n (BM n ) and their principal divisors, that will be useful in the following. Let Proof. Define t := x + y − 1, so thatθ 0 = −t/(x + y) andθ 0 has principal divisor (θ 0 ) = (t) − (x + y). Let P 1 , . . . , P q+1 be the q + 1 points at infinity of the Hermitian curve H q , namely P i := (1 : a i : 0) in homogeneous coordinates (x : y : w), with a q+1 i = 1. The line defined by t intersects H q in exactly q + 1 distinct points; these points are Q b := (1 − b : b : 1) for b ∈ F q 2 satisfying b q + b = 0 and P 1 . Then, the principal divisor of t in F q 2n (H q ) is: For all b ∈ F q 2 satisfying b q + b = 0 let Q i b := (1 − b : b : 0 : 1) be the unique point in O 2 lying over Q b and note that Q 1 = Q i 0 . Further, P i is the unique point in O 1 lying over P i for all i = 1, . . . , q + 1. Then, the principal divisor of t in F q 2n (BM n ) is Hence, We are now ready to focus on the two-point Weierstrass semigroup H(Q 1 , P 1 ). To this aim, we first give an explicit description of the ring of functions that are regular outside Q 1 and P 1 and we compute the period of H(Q 1 , P 1 ).
Proof. Assume by contradiction that k(Q 1 − P 1 ) is a principal divisor for some k ∈ {1, . . . , q n }. Let f ∈ F q 2n (BM n ) such that (f ) = k(Q 1 − P 1 ). In particular k is a non-gap of the Weierstrass semigroup H(Q 1 ), as v Q 1 (f −1 ) = −k and Q 1 is the only pole of f −1 . The smallest non-zero element of H(Q 1 ) is q n + 1 − m (see (5)), hence q n + 1 − m ≤ k ≤ q n and we can write then m − j must be a non-gap of the Weierstrass semigroup H(Q 1 ); this is not possible, as 0 < m − j < qm = q n + 1 − m.
Given Proposition 3.3 and Lemma 3.4, we are now able to prove the main theorem of the paper, which provides the explicit expression of the function τ Q 1 ,P 1 . Note that the knowledge of the function τ Q 1 ,P 1 is sufficient to determine the two-point Weierstrass semigroup H(Q 1 , P 1 ).
Proof. Define the mapτ : Z → Z such thatτ (i) = k(q n + 1) + (γ + )mq + β(q 2 − q) for all i ∈ Z and k, , β, γ as in the assumptions. We will prove that τ (i) = τ Q,P (i) for all i ∈ Z. In the following, we fix i ∈ Z, so that k, , β, γ are fixed too. Choose M non-negative integers i 1 , . . . , i M such that From (7), (8) and Lemma 3.2, the principal divisor of f is where E and E are effective divisors. The above computation shows that (i,τ (i)) belongs to H(Q 1 , P 1 ) and thusτ (i) ≥ τ Q 1 ,P 1 (i) by definition of τ Q 1 ,P 1 . Finally, we can use Lemma 2.6 d) to show that the equalityτ (i) = τ Q 1 ,P 1 (i) holds. In fact, we have just proved thatτ (i) ≥ τ Q 1 ,P 1 (i) for all i ∈ Z and therefore π+c−1 i=c (i +τ (i)) ≥ π+c−1 i=c (i + τ Q,P (i)) = πg (10) for all c ∈ Z. To conclude, it is enough to check that the left side of equation (10) is equal to πg. We can choose c = −π + 1 without loss of generality, so that Writing γ = 1 M (β + (M − β) mod M ), the quantity on the right side of equation (11) yields It can be checked with a direct computation that the above quantity is equal to 1 2 (q n + 1)(q − 1)(q n+1 + q n − q 2 ) = πg.

Computation of the order bound and results
We are now ready to compute the order bound for dual codes of two-point AG codes from the BM n curve, for all n ≥ 3 odd. Let G = aQ 1 + bP 1 and denote with δ := a + b its degree. We define the divisor D to be the sum of all the F q 2n -rational points of BM n different from Q 1 and P 1 . The degree of D is therefore deg(D) = N n − 2, where N n = q 2n+2 − q n+3 + q n+2 + 1 is the number of F q 2n -rational points of BM n . The two-point AG code C L (D, G) and its dual C L (D, G) ⊥ are linear subspaces of F Nn−2 q 2n . If δ ≥ N n + 2g − 3, then the code C L (D, G) ⊥ is the zero code; this follows from the fact that, if δ ≥ N n + 2g − 3, the divisors G and G − D are nonspecial, as their degrees exceed 2g − 2 and, from the Riemann-Roch theorem, dim(C L (D, G)) = dim(L(G)) − dim(L(G − D)) = N n − 2.
Define ∆ := 4g − 1. As pointed out in Section 2, it is sufficient to determine the order bound for the code C L (D, G) ⊥ for the case δ < ∆ only, since the order bound coincides with the Goppa bound if the degree of G is larger than or equal to ∆ (see Lemma 2.3). The condition δ < ∆, which also implies deg(G) < deg(D), makes the determination of the dimension of C L (D, G) ⊥ a particularly easy task; in fact dim(C L (D, G) ⊥ ) = N n − 2 − dim(L(G)).
The dimension of L(G) can be conveniently computed applying Theorem 2.8 with the map τ Q 1 ,P 1 defined in Theorem 3.5.
We also compared our results obtained using Algorithm 4.1 with the results obtained using [1, Algorithm 1] for q = 2 and n = 5. In this case the two curves BM 5 and GGS 5 are not isomorphic. Table 2 Table 2: For q = 2, n = 5, Table 2 reports the largest estimate for the minimum distance d of a code C L (D, aQ 1 + bP 1 ) ⊥ of length N 5 − 2 = 3967 and dimension k and compares d with d 2 , the largest estimate obtained in [1] for codes of same length and dimension. Only the cases where d > d 2 are reported.