AG codes from Fq7\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\mathbb{F}}_{q^7}}$$\end{document}-rational points of the GK maximal curve

In Beelen and Montanucci (Finite Fields Appl 52:10–29, 2018) and Giulietti and Korchmáros (Math Ann 343:229–245, 2009), Weierstrass semigroups at points of the Giulietti–Korchmáros curve X\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {X}}$$\end{document} were investigated and the sets of minimal generators were determined for all points in X(Fq2)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {X}}(\mathbb {F}_{q^2})$$\end{document} and X(Fq6)\X(Fq2)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {X}}(\mathbb {F}_{q^6})\setminus {\mathcal {X}}( \mathbb {F}_{q^2})$$\end{document}. This paper completes their work by settling the remaining cases, that is, for points in X(F¯q)\X(Fq6)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {X}}(\overline{\mathbb {F}}_{q}){\setminus }{\mathcal {X}}( \mathbb {F}_{q^6})$$\end{document}. As an application to AG codes, we determine the dimensions and the lengths of duals of one-point codes from a point in X(Fq7)\X(Fq)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {X}}(\mathbb {F}_{q^7}){\setminus }{\mathcal {X}}( \mathbb {F}_{q})$$\end{document} and we give a bound on the Feng–Rao minimum distance dORD\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$d_{ORD}$$\end{document}. For q=3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$q=3$$\end{document} we provide a table that also reports the exact values of dORD\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$d_{ORD}$$\end{document}. As a further application we construct quantum codes from Fq7\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {F}_{q^7}$$\end{document}-rational points of the GK-curve.


Introduction
Algebraic geometric methods have largely been used for the construction of error-correcting linear codes from algebraic curves. The essential idea going back to Goppa's work (see [10] and [11]) is that a linear code can be obtained from an algebraic curve X defined over a finite field q by evaluating certain rational functions whose poles are prescribed by a given q -rational divisor G at some q -rational divisor D whose support is disjoint from that of G. These codes are 1 3 called functional (or evaluation) codes. The dual of such a code can also be obtained by using Goppa's idea, taking residues of differential forms rather than rational functions. They are called differential AG codes. Actually, any linear code is an AG code; see [19].
AG codes are proven to have good performances provided that X , G and D are carefully chosen in an appropriate way. In particular, AG codes with better parameters can arise from curves which have many q -rational points, especially from maximal curves which are curves defined over q with q square whose number of q -rational points X( q ) attains the Hasse-Weil upper bound, namely �X( q )� = q + 1 + 2 √ q , where is the genus of X ; for AG codes from maximal curves see for instance [6,13,17,18]. Regarding the choice of the two divisors D and G, the latter is typically taken to be a multiple mP of a single point P of degree one. Such codes are known as one-point codes, and have been extensively investigated; see for instance [5,8,15,21,24].
An important ingredient for the construction of one-point AG codes is the Weierstrass semigroup H(P) of X at P, whose elements are the non-negative integers k for which there exists a rational function on X having pole divisor kP. Indeed, the knowledge of this semigroup allows to obtain useful information on the parameters of functional and differential codes. Although the structure of H(P) is not always the same for every point P of X , it is known that this holds true for all but a finite number of points P ∈ X . A point for which the Weierstrass semigroup is not the typical one is a called a Weierstrass point. If G(P) ∶= ℕ⧵H(P) denotes the set of gaps at P, it is well known that the size of G(P) equals the genus of X for every P ∈ X ; see [22,Theorem 1.6.8].
Several papers have been dedicated to the construction of AG codes from the GK curves; see [1,2,4,7]. The GK-curves are q 6-maximal curves due to Giulietti and Korchmáros, which provided the first family of maximal curves that are not subcovers of the Hermitian curve [9]. The Weierstrass semigroup is known at any q 2-rational point of the GK curve X , see [9], as well as at any point in X( q 6 )⧵X( q 2 ) , see [3]. In the latter paper, see Result 7, the authors also deal with Weierstrass semigroups at points in X( q )⧵X( q 6 ) , showing that the Weierstrass points of the GK curve are exactly its q 6-rational points. However the problem of determining the generators of a Weierstrass semigroup H(P) with P ∈ X( q )⧵X( q 6 ) has remained open. In the present paper we solve this problem. Therefore the Weierstrass semigroups at the points of the GK curve are completely determined.
Then, our main result is the following theorem.
Theorem 1 Let X be the GK curve over q and let P ∈ X(̄ q )⧵X( q 6 ).

3
AG codes from q 7-rational points of the GK maximal curve This theorem together with the already quoted previous results provide a complete description of the Weierstrass semigroups at any point of the GK-curve. Theorem 2 Let X be the GK curve over q and P be a point of X. Then one of the following occurs, where e(H(P)) denotes the number of generators of H(P).
-P ∈ X( q 2 ) , H(P) = ⟨q 3 − q 2 + q, q 3 , q 3 + 1⟩ and e(H(P)) = 3; and e(H(P)) = q + 2; -q > 2 , P ∈ X(̄ q )⧵X( q 6 ) , H(P) = ⟨S⟩ and e(H(P)) = 2q 2 − q; -q = 2 , P ∈ X(̄ q )⧵X( q 6 ) , H(P) = ⟨7, 8, 12, 13, 18⟩ and e(H(P)) = 5, The above results are then applied to the construction of AG codes and quantum codes from an q 7-rational point of the GK curve. More in detail, Sect. 4 is devoted to the construction of dual codes of one-point AG codes. We investigate their parameters and we provide explicit tables in the case q = 3 . In Sect. 5, by applying the CSS construction to the codes constructed in Sect. 4, we exhibit families of quantum codes. Also in this case, explicit tables are provided.

Numerical semigroups
A subset H of ℕ 0 containing 0, which is closed under sums and which has finite complement is called a numerical semigroup. The main reference for the theory of numerical semigroups is [20]. Associated to H there are several invariants, parameters and subsets, the most important being the genus g(H) and the gapset G(H) = ℕ 0 ⧵H . The genus is the cardinality of the gapset, which, by definition, is finite.
For a nonempty subset A = {a 1 , … , a n } of ℕ 0 , ⟨A⟩ denotes the smallest subset of ℕ 0 containing A, 0 and closed under addition; clearly ⟨A⟩ = ℕ 0 a 1 + ⋯ + ℕ 0 a n . For a numerical semigroup H, the minimal system of generators {h 1 , … , h e } is the smallest subset of H such that H = ⟨h 1 , … , h e ⟩ , and its cardinality e(H) is called the embedding dimension of H. Definition 1 For a numerical semigroup H and n ∈ H⧵{0} , the Apéry set of n is A strong connection between the Apéry set and the genus is given by the following result.

Weierstrass semigroups and AG codes
For a curve X , we adopt the usual notation and terminology. In particular, q (X) and X( q ) denote the field of q -rational functions on X and the set of q -rational points of X , respectively, and Div(X) denotes the set of divisors of X , where a divisor D ∈ Div(X) is a formal sum n 1 P 1 + ⋯ + n r P r , with P i ∈ X , n i ∈ ℤ and P i ≠ P j if i ≠ j . The support Supp (D) of the divisor D is the set of points P i such that . For a function f ∈ q (X) , (f), (f ) 0 and (f ) ∞ are the divisor of f, its divisor of zeroes and its divisor of poles, respectively. The Weierstrass semigroup H(P) at P ∈ X is The Riemann-Roch space associated with an q -rational divisor D is and its vector space dimension over q is (D).
Fix a set of pairwise distinct q -rational points {P 1 , ⋯ , P N } , and let D = P 1 + ⋯ + P N . Take another divisor G whose support is disjoint from the support of D.
The dual code C ⊥ (D, G) can be obtained in a similar way from the q (X)-vector space (X) of differential forms over X . With ∈ (X) , there is associated the divisor ( ) of X , and for an q -rational divisor D, is a q -vector space of rational differential forms over X . Then the code C ⊥ (D, G) coincides with the (linear) subspace of N q which is the image of the vector space (G − D) under the linear map res D ∶ (G − D) ↦ N q given by res D ( ) = (res P 1 ( ), … , res P N ( )) , where res P i ( ) is the residue of at P i . In par- In the case where G = P , ∈ ℕ 0 , P ∈ X( q ) , the AG code C (D, G) is referred to as one-point AG code. For a Weierstrass semigroup H(P) = { 0 = 0 < 1 < 2 < ⋯} and an integer ≥ 0 , the Feng-Rao function is Consider with P, P 1 , … , P N pairwise distint points in X( q ) . The number is a lower bound for the minimum distance d(C (P)) of the code C (P) which is called the order bound or the Feng-Rao designed minimum distance of C (P) ; see [12,Theorem 4.13].

The GK curve
Let q be a prime power and =̄ q . The Giulietti-Korchmáros (GK) curve X is the first example of a q 6-maximal curve which is covered by the Hermitian curve over q 6 only for q = 2 ; see [9]. The GK curve X is a non-singular curve, viewed as curve of PG (3, ) , defined by the affine equations It has genus (X) = 1 2 (q 5 − 2q 3 + q 2 ) and as many as q 8 − q 6 + q 5 + 1 q 6-rational points. From Eq. (2), the GK curve is a Galois extension (in fact a Kummer extension) of the Hermitian curve H q over q 2 given by the affine equation Y q+1 = X q + X . The automorphism group Aut(X) of X is also defined over q 6 . It has order q 3 (q 3 + 1)(q 2 − 1)(q 2 − q + 1) and contains a normal subgroup isomorphic to SU (3, q).
The set of q 6-rational points of X splits into two orbits O 1 = X( q 2 ) and O 2 = X( q 6 )⧵X( q 2 ) under the action of Aut(X) . The orbit O 1 is non-tame and has size q 3 + 1 , whereas O 2 is tame of size q 3 (q 3 + 1)(q 2 − 1) . Furthermore, these are the only short orbits of Aut(X) , and Aut(X) acts on O 1 as PGU (3, q) in its doubly transitive permutation representation; see [9,Theorem 7]. As it is known, the structure of Weierstrass semigroups is invariant under the action of automorphism groups; see [22,Lemma 3.5.2].
In Sect. 4 we will construct AG codes from q 7-rational points of the GK curve. In order to compute the number of those points the following results will be useful.
Result 5 [16, Propositions 1 and 2] Let X be a curve defined over q . Then the following holds.
1. if X is q -maximal and n is odd, then X is q n-maximal; 2. if X is q 2n-maximal, then |X( q n )| = q n + 1.
As the Hermitian curve H q is q 2-maximal, the following corollary of Result 5 holds.

Result 6
If d is odd, the number of q d-rational points of the Hermitian curve H q is q d + 1.
, and hence q 2 − q + 1 and q 7 − 1 are coprime. Therefore, the equation X q 2 −q+1 = c , with c ∈ q 7 , has exactly one solution. This shows that the number of q 7-rational points of X equals the number of q 7-rational points of the Hermitian curve H q . Therefore the claim follows by Result 6. ◻ In [3] the Weierstrass semigroup H(P) for P ∈ X( q )⧵X( q 6 ) was studied. In particular, the authors showed that H(P) is the same for every P ∈ X( q )⧵X( q 6 ) , and computed explicitly the set of gaps G(P) = ℕ 0 ⧵H(P).

Result 7 [3, Theorem 4.10]
Let P be a point of X with P ∈ X( q )⧵X( q 6 ) . Then the set of gaps at P is Each element of G(P) admits a unique representation as in (3), i.e. each element of G(P) is uniquely identified by the tuple of coefficients (i, j, k, m, n 1 , … , n q−2 ) . Furthermore the set G(P) is the disjoint union of the sets G 1 , G 2 , G 3 , where -G 1 is the subset of G(P) corresponding to the coefficients (i, 0, k, m, 0, … , 0) .

3
AG codes from q 7-rational points of the GK maximal curve

Proof of Theorem 1
For q = 2 the claim is already known; see [3,Example 4.12]. Therefore, assume q > 2 and let T denote the semigroup generated by S. To show T = H(P) it is enough to prove that T ⊂ H(P) and that T and H(P) have the same genus. For this purpose, some properties of the following subsets of T are useful.
Proof Let x a,i,j denote the element of Ap 1 corresponding to the choices of the parameters a, i, j, that is We use an analogous notation for the elements of Ap 2,1 , Ap 2,2 , Ap 3 and Ap 4 .
-Let x a,i,j ∈ Ap 1 and x̄i ,j ∈ Ap 3 . If x a,i,j = x̄i ,j then that modulo q yields a = 1 , a contradiction with a ≥ 2.
-Ap 3 ∩ Ap 4 is empty since for every element x of Ap 3 , x + 1 is divisible by q, but this fails for any element of Ap 4 .
Proposition 3 The cardinalities of the sets Ap 1 , Ap 2 , Ap 3 , Ap 4 are as follows Proof From the definition of Ap 1 , Ap 2,1 , Ap 2,2 Ap 3 , and Ap 4 , a straightforward computation shows that different choices of the parameters lead to different elements in the corresponding set.
We provide here the proof for the case Ap 1 . Analogous computations can be applied to the other cases. Let x, y ∈ Ap 1 , so Assume that x = y holds. Then a ≡ā (mod q) , and since a,ā ∈ {2, … , q − 1} , we obtain a =ā . Therefore whence By applying the same argument as above, we obtain i =ī . Finally, this implies j =j , and so the claim follows. ◻

Proposition 4 If x ∈ A then x − q 3 ∉ H(P).
Proof For each element x in A, we exhibit a representation of x − q 3 as in (3). The claim trivially holds for x = 0 . Moreover, AG codes from q 7-rational points of the GK maximal curve Therefore x − q 3 ∉ H(P) by (3).
◻ We use Proposition 4 to prove the following lemma.

Lemma 1 The semigroup T is contained in H(P).
Proof Since T = ⟨S⟩ , it suffices to show that S = S 1 ∪ S 2 ⊆ H(P) . We carry out the computation for the case x ∈ S 1 = Ap 2,1 . Analogous computation can be done for the other elements in S 2 = Ap 2,2 . Take x ∈ S 1 . Then for some 0 ≤ i ≤ q − 1 and 0 ≤ j ≤ q − 1 . It may be observed that We assume on the contrary x ∈ G(P) . Taking into account Result 7 we distinguish three cases according to either x ∈ G 1 , or x ∈ G 2 , or x ∈ G 3 .

AG codes from q 7-rational points of the GK curve
In this section we construct a family of AG codes from q 7-rational points of the GK curve. For q = 3 the parameters of the codes obtained are reported in the table below.
We keep our notation in Sect. 2.2. In particular, for a point P ∈ X( q 7 )⧵X( q ) , H(P) = {0 = 1 < 2 < ...} denotes the Weierstrass semigroup at P and C (P) stands for the dual code C (P) = C ⟂ (D, P) , where is a divisor supported at all q 7-rational points of X but P. From the Feng-Rao lower bound on the minimum distance of C (P) , we have that C (P) is an [n, k, d] q 7 linear code, with n = q 7 , k = n − and where d * = deg(G) − 2 + 2 denotes the designed minimum distance of C (P) . We remark that the Feng-Rao lower bound can be computed only in terms of the Weierstrass semigroup H(P), that we explicitly described in Theorem 1.
As a consequence of Results 4 and 8 the following result follows.

Quantum codes from q 7-rational points of the GK curve
It is known that quantum codes can be constructed from (classical) linear codes by using the so-called CSS construction; see [14,Lemma 2.5]. Our aim is to show how the CSS-construction applies to one-point AG codes on the GK curve. As before q is a prime power. Let ℍ = (ℂ q ) ⊗n = ℂ q ⊗ ⋯ ⊗ ℂ q be a q n -dimensional Hilbert space. Then the q-ary quantum code C of length n and dimension k are the q k -dimensional Hilbert subspace of ℍ . Such quantum codes are denoted by [[n, k, d]] q , where d is the minimum distance. As in the ordinary case, C can correct up to ⌊ d−1 2 ⌋ errors. Moreover, the quantum version of the Singleton bound states that for a [[n, k, d]] q -quantum code, 2d + k ≤ 2 + n holds. Again, by analogy with the ordinary case, the quantum Singleton defect and the relative quantum Singleton defect are defined to be Q ∶= n − k − 2d + 2 and Q ∶= Q n , respectively. We recall [ Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article's Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http:// creat iveco mmons. org/ licen ses/ by/4. 0/.