1 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 \({\mathcal {X}}\) defined over a finite field \(\mathbb {F}_q\) by evaluating certain rational functions whose poles are prescribed by a given \(\mathbb {F}_q\)-rational divisor G at some \(\mathbb {F}_q\)-rational divisor D whose support is disjoint from that of G. These codes are 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 \(\mathcal {X}\), G and D are carefully chosen in an appropriate way. In particular, AG codes with better parameters can arise from curves which have many \(\mathbb {F}_q\)-rational points, especially from maximal curves which are curves defined over \(\mathbb {F}_q\) with q square whose number of \(\mathbb {F}_q\)-rational points \({\mathcal {X}}(\mathbb {F}_q)\) attains the Hasse-Weil upper bound, namely \(|{\mathcal {X}}(\mathbb {F}_q)| = q+1+2\mathfrak {g} \sqrt{q}\), where \(\mathfrak {g}\) is the genus of \({\mathcal {X}}\); 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 \({\mathcal {X}}\) at P, whose elements are the non-negative integers k for which there exists a rational function on \({\mathcal {X}}\) 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 \({\mathcal {X}}\), it is known that this holds true for all but a finite number of points \(P\in {\mathcal {X}}\). A point for which the Weierstrass semigroup is not the typical one is a called a Weierstrass point. If \(G(P):=\mathbb {N}{\setminus } H(P)\) denotes the set of gaps at P, it is well known that the size of G(P) equals the genus \(\mathfrak {g}\) of \({\mathcal {X}}\) for every \(P\in {\mathcal {X}}\); 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 \(\mathbb {F}_{q^6}\)-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 \(\mathbb {F}_{q^2}\)-rational point of the GK curve \({\mathcal {X}}\), see [9], as well as at any point in \({\mathcal {X}}(\mathbb {F}_{q^6}){\setminus }{\mathcal {X}}(\mathbb {F}_{q^2})\), see [3]. In the latter paper, see Result 7, the authors also deal with Weierstrass semigroups at points in \({\mathcal {X}}(\overline{\mathbb {F}}_q){\setminus } {\mathcal {X}}(\mathbb {F}_{q^6})\), showing that the Weierstrass points of the GK curve are exactly its \(\mathbb {F}_{q^6}\)-rational points. However the problem of determining the generators of a Weierstrass semigroup H(P) with \(P\in {\mathcal {X}}(\overline{\mathbb {F}}_q){\setminus } {\mathcal {X}}(\mathbb {F}_{q^6})\) 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.

Let \(S=S_1\cup S_2\), with

$$\begin{aligned} \begin{array}{llll} S_1 &{}=&{}\{q^3+i(q^3-q)+j(q^4-q^3-q^2)\,|\, i=0,\ldots ,q-1, \quad j=0,\ldots ,q-1 \},\\ S_2 &{}=&{} \{q^3-1+i(q^3-q)+j(q^4-q^2-1)\,|\, i=0,\ldots ,q-1, \quad j=0,\ldots ,q-2 \}. \end{array} \end{aligned}$$

Then, our main result is the following theorem.

Theorem 1

Let \({\mathcal {X}}\) be the GK curve over \(\mathbb {F}_q\) and let \(P\in \mathcal {X}(\bar{\mathbb {F}}_q){\setminus }\mathcal {X}(\mathbb {F}_{q^6})\). Then if \(q>2\), S is a minimal set of generators for the Weierstrass semigroup H(P). For \(q=2\), the minimal set of generators for H(P) is \(\{7, 8, 12, 13, 18\}\).

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 \({\mathcal {X}}\) be the GK curve over \(\mathbb {F}_q\) and P be a point of \({\mathcal {X}}\). Then one of the following occurs, where e(H(P)) denotes the number of generators of H(P).

  • \(P\in {\mathcal {X}}(\mathbb {F}_{q^2})\), \(H(P)=\langle q^3 - q^2 + q, q^3, q^3 + 1\rangle\) and \(e(H(P))=3\);

  • \(P\in {\mathcal {X}}(\mathbb {F}_{q^6}){\setminus }{\mathcal {X}}(\mathbb {F}_{q^2})\), \(H(P)=\langle q^3 - q + 1, q^3 + 1, q^3 + i(q^4 - q^3 - q^2 + q - 1):\, i = 0, \dots , q - 1\rangle\) and \(e(H(P))=q+2\);

  • \(q>2\), \(P\in \mathcal {X}(\bar{\mathbb {F}}_q){\setminus }\mathcal {X}(\mathbb {F}_{q^6})\), \(H(P)=\langle S\rangle\) and \(e(H(P))=2q^2-q\);

  • \(q=2\), \(P\in \mathcal {X}(\bar{\mathbb {F}}_q){\setminus }\mathcal {X}(\mathbb {F}_{q^6})\), \(H(P)=\langle 7, 8, 12, 13, 18\rangle\) and \(e(H(P))=5\),

The above results are then applied to the construction of AG codes and quantum codes from an \(\mathbb {F}_{q^7}\)-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.

2 Background on numerical semigroups and on the GK-curve

2.1 Numerical semigroups

A subset H of \(\mathbb {N}_0\) 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 \(G(H)=\mathbb {N}_0{\setminus } H\). The genus is the cardinality of the gapset, which, by definition, is finite.

For a nonempty subset \(A =\{ a_1,\dots , a_n\}\) of \(\mathbb {N}_0\), \(\langle A\rangle\) denotes the smallest subset of \(\mathbb {N}_0\) containing A, 0 and closed under addition; clearly \(\langle A\rangle =\mathbb {N}_0a_1 + \cdots + \mathbb {N}_0a_n\). For a numerical semigroup H, the minimal system of generators \(\{h_1,\dots ,h_e\}\) is the smallest subset of H such that \(H=\langle h_1,\dots ,h_e\rangle\), and its cardinality e(H) is called the embedding dimension of H.

Definition 1

For a numerical semigroup H and \(n\in H{\setminus } \{0\}\), the Apéry set of n is

$$\begin{aligned} Ap(H,n):=\{x \in H \,|\,x-n \not \in H\}. \end{aligned}$$

A strong connection between the Apéry set and the genus is given by the following result.

Result 3

[20, Lemma 2.4, Proposition 2.12] Let H be a numerical semigroup and n a nonzero element of H. Then \(|Ap(H,n)|=n\) and

$$\begin{aligned} g(H)=\frac{1}{n}\sum _{x\in Ap(H,n)} x- \frac{n-1}{2}. \end{aligned}$$
(1)

2.2 Weierstrass semigroups and AG codes

For a curve \(\mathcal {X}\), we adopt the usual notation and terminology. In particular, \(\mathbb {F}_q(\mathcal {X})\) and \(\mathcal {X}(\mathbb {F}_q)\) denote the field of \(\mathbb {F}_q\)-rational functions on \({\mathcal {X}}\) and the set of \(\mathbb {F}_q\)-rational points of \(\mathcal {X}\), respectively, and \(\mathrm{{Div}}(\mathcal {X})\) denotes the set of divisors of \(\mathcal {X}\), where a divisor \(D\in \mathrm{{Div}}(\mathcal {X})\) is a formal sum \(n_1P_1+\cdots +n_rP_r\), with \(P_i \in \mathcal {X}\), \(n_i \in \mathbb {Z}\) and \(P_i\ne P_j\) if \(i\ne j\). The support \(\text{ Supp }(D)\) of the divisor D is the set of points \(P_i\) such that \(n_i\ne 0\), while \(\deg (D)=\sum _i n_i\) is the degree of D. The divisor D is \({\mathbb {F}}_q\)-rational if \(n_i\ne 0\) implies \(P_i\in {\mathcal {X}}(\mathbb {F}_q)\). For a function \(f \in \mathbb {F}_q(\mathcal {X})\), (f), \((f)_0\) and \((f)_{\infty }\) are the divisor of f, its divisor of zeroes and its divisor of poles, respectively. The Weierstrass semigroup H(P) at \(P\in {\mathcal {X}}\) is

$$\begin{aligned} H(P) := \{n \in \mathbb {N}_0 \ | \ \exists f \in \mathbb {F}_q(\mathcal {X}), (f)_{\infty }=nP\}= \{\rho _0=0<\rho _1<\rho _2<\cdots \}. \end{aligned}$$

The Riemann-Roch space associated with an \({\mathbb {F}}_q\)-rational divisor D is

$$\begin{aligned} \mathcal {L}(D) := \{ f \in \mathcal {X}(\mathbb {F}_q) \ : \ (f)+D \ge 0\}\cup \{0\} \end{aligned}$$

and its vector space dimension over \(\mathbb {F}_q\) is \(\ell (D)\).

Fix a set of pairwise distinct \(\mathbb {F}_q\)-rational points \(\{P_1,\cdots ,P_N\}\), and let \(D=P_1+\cdots +P_N\). Take another divisor G whose support is disjoint from the support of D. The AG code C(D, G) is the (linear) subspace of \(\mathbb {F}_q^N\) which is defined as the image of the evaluation map \(ev : \mathcal {L}(G) \rightarrow \mathbb {F}_q^N\) given by \(ev(f) = (f(P_1),f(P_2) ,\ldots ,f(P_N))\). In particular C(D, G) has length N. Moreover, if \(N>\deg (G)\) then ev is an embedding and \(\ell (G)\) equals the dimension of C(D, G). The minimum distance d of C(D, G), usually depends on the choice of D and G. A lower bound for d is \(d^*=N-\deg (G)\), where \(d^*\) is called the Goppa designed minimum distance of C(D, G). Furthermore, if \(\deg (G)>2\mathfrak {g}-2\) then \(k=\deg (G)-\mathfrak {g}+1\) by the Riemann--Roch Theorem; see [12, Theorem 2.65].

The dual code \(C^{\bot } (D,G)\) can be obtained in a similar way from the \(\mathbb {F}_q({\mathcal {X}})\)-vector space \(\varOmega ({\mathcal {X}})\) of differential forms over \({\mathcal {X}}\). With \(\omega \in \varOmega ({\mathcal {X}})\), there is associated the divisor \((\omega )\) of \({\mathcal {X}}\), and for an \(\mathbb {F}_q\)-rational divisor D,

$$\begin{aligned} \varOmega (D):=\{\omega \in \varOmega ({\mathcal {X}})\ :\ (\omega )\ge D\}\cup \{0\} \end{aligned}$$

is a \(\mathbb {F}_q\)-vector space of rational differential forms over \({\mathcal {X}}\). Then the code \(C^{\bot }(D,G)\) coincides with the (linear) subspace of \(\mathbb {F}_q^N\) which is the image of the vector space \(\varOmega (G-D)\) under the linear map \(res_D:\varOmega (G-D)\mapsto \mathbb {F}_q^N\) given by \(res_D(\omega )=(res_{P_1}(\omega ),\dots ,res_{P_N}(\omega ))\), where \(res_{P_i}(\omega )\) is the residue of \(\omega\) at \(P_i\). In particular, \(C^{\bot }(D,G)\) is an AG code with dimension \(k^{\bot }=N-k\) and minimum distance \(d^{\bot }\ge \deg {(G)}-2\mathfrak {g}+2\).

In the case where \(G=\alpha P\), \(\alpha \in \mathbb {N}_0\), \(P \in \mathcal {X}(\mathbb {F}_q)\), the AG code C (D, G) is referred to as one-point AG code. For a Weierstrass semigroup \(H(P)= \{\rho _0=0<\rho _1<\rho _2<\cdots \}\) and an integer \(\ell \ge 0\), the Feng–Rao function is

$$\begin{aligned} \nu _\ell := | \{(i,j) \in \mathbb {N}_0^2 \ : \ \rho _i+\rho _j = \rho _{\ell +1}\}|. \end{aligned}$$

Consider

$$\begin{aligned} {C}_{\ell }(P)= {C}^{\bot }(P_1+P_2+\cdots +P_N,\rho _{\ell }P), \end{aligned}$$

with \(P,P_1,\ldots ,P_N\) pairwise distint points in \(\mathcal {X}(\mathbb {F}_q)\). The number

$$\begin{aligned} d_{ORD} ({C}_{\ell }(P)) := \min \{\nu _{m} \ : \ m \ge \ell \} \end{aligned}$$

is a lower bound for the minimum distance \(d({C}_{\ell }(P))\) of the code \({C}_{\ell }(P)\) which is called the order bound or the Feng–Rao designed minimum distance of \({C}_{\ell }(P)\); see [12, Theorem 4.13].

For the following result see [12, Theorem 5.24].

Result 4

\(d_{ORD} ({C}_{\ell }(P))\ge \ell +1-\mathfrak {g}\). Equality holds if \(\ell \ge 2c-\mathfrak {g}-1\) with \(c=\max \{m \in \mathbb {Z} \ : \ m-1 \notin H(P)\}.\)

2.3 The GK curve

Let q be a prime power and \(\mathbb {K} = \bar{\mathbb {F}}_q\). The Giulietti-Korchmáros (GK) curve \({\mathcal {X}}\) is the first example of a \(\mathbb {F}_{q^6}\)-maximal curve which is covered by the Hermitian curve over \(\mathbb {F}_{q^6}\) only for \(q=2\); see [9]. The GK curve \({\mathcal {X}}\) is a non-singular curve, viewed as curve of \(PG(3,\mathbb {K})\), defined by the affine equations

$$\begin{aligned} \left\{ \begin{array}{ll} Y^{q+1}=X^q+X, &{} \\ Z^{q^2-q+1}=Y^{q^2}-Y. &{} \end{array} \right. \end{aligned}$$
(2)

It has genus \(\mathfrak {g}({\mathcal {X}}) =\frac{1}{2} (q^5 - 2q^3 + q^2)\) and as many as \(q^8 - q^6 + q^5 + 1\) \(\mathbb {F}_{q^6}\)-rational points. From Eq. (2), the GK curve is a Galois extension (in fact a Kummer extension) of the Hermitian curve \(\mathcal {H}_q\) over \(\mathbb {F}_{q^2}\) given by the affine equation \(Y^{q+1} = X^{q} + X\). The automorphism group \(\mathrm{Aut}({\mathcal {X}})\) of \({\mathcal {X}}\) is also defined over \(\mathbb {F}_{q^6}\). 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 \(\mathbb {F}_{q^6}\)-rational points of \({\mathcal {X}}\) splits into two orbits \(\mathcal {O}_1={\mathcal {X}}(\mathbb {F}_{q^2})\) and \(\mathcal {O}_2={\mathcal {X}}(\mathbb {F}_{q^6}){\setminus }{\mathcal {X}}(\mathbb {F}_{q^2})\) under the action of \(\mathrm{Aut}({\mathcal {X}})\). The orbit \(\mathcal {O}_1\) is non-tame and has size \(q^3 + 1\), whereas \(\mathcal {O}_2\) is tame of size \(q^3(q^3 + 1)(q^2-1)\). Furthermore, these are the only short orbits of \(\mathrm{Aut}({\mathcal {X}})\), and \(\mathrm{Aut}({\mathcal {X}})\) acts on \(\mathcal {O}_1\) as \({\text{ PGU }}(3, q)\) 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 \(\mathbb {F}_{q^7}\)-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 \({\mathcal {X}}\) be a curve defined over \(\mathbb {F}_q\). Then the following holds.

  1. 1.

    if \({\mathcal {X}}\) is \(\mathbb {F}_{q}\)-maximal and n is odd, then \({\mathcal {X}}\) is \(\mathbb {F}_{q^{n}}\)-maximal;

  2. 2.

    if \({\mathcal {X}}\) is \(\mathbb {F}_{q^{2n}}\)-maximal, then \(|{\mathcal {X}}(\mathbb {F}_{q^n})|=q^n + 1\).

As the Hermitian curve \(\mathcal {H}_q\) is \(\mathbb {F}_{q^2}\)-maximal, the following corollary of Result 5 holds.

Result 6

If d is odd, the number of \(\mathbb {F}_{q^d}\)-rational points of the Hermitian curve \(\mathcal {H}_q\) is \(q^d+1\).

Proposition 1

\(|\mathcal {X}(\mathbb {F}_{q^7})|=q^7+1\).

Proof

Observe that \((q^7-1,q^2-q+1)=(q^7-1-(q^5+q^4-q^2-q)(q^2-q+1),q^2-q+1)=(q-1,q^2-q+1)=1\), and hence \(q^2-q+1\) and \(q^7-1\) are coprime. Therefore, the equation \(X^{q^2-q+1}=c\), with \(c\in \mathbb {F}_{q^7}\), has exactly one solution. This shows that the number of \(\mathbb {F}_{q^7}\)-rational points of \(\mathcal {X}\) equals the number of \(\mathbb {F}_{q^7}\)-rational points of the Hermitian curve \(\mathcal {H}_q\). Therefore the claim follows by Result 6. \(\square\)

In [3] the Weierstrass semigroup H(P) for \(P\in \mathcal {X}(\overline{\mathbb {F}}_q){\setminus } \mathcal {X}(\mathbb {F}_{q^6})\) was studied. In particular, the authors showed that H(P) is the same for every \(P\in \mathcal {X}(\overline{\mathbb {F}}_q){\setminus } \mathcal {X}(\mathbb {F}_{q^6})\), and computed explicitly the set of gaps \(G(P)=\mathbb {N}_0{\setminus } H(P)\).

Result 7

[3, Theorem 4.10] Let P be a point of \({\mathcal {X}}\) with \(P\in \mathcal {X}(\overline{\mathbb {F}}_q){\setminus } \mathcal {X}(\mathbb {F}_{q^6})\). Then the set of gaps at P is

$$\begin{aligned} \begin{aligned} G(P)&= \{ iq^3 + kq + m(q^2 + 1) + \sum ^{q-2}_{s=1} ( n_s(s + 1)q^2) \\&\quad +\, j + 1 \,\mid \, i, j, k,m,\ldots , n_{q-2} \ge 0, \\ j&\le q-1,\text { and }i + j + k + mq + \sum ^{q-2}_{s=1} (n_s((s + 1)q - s)) \le q^2 - 2\}. \end{aligned} \end{aligned}$$
(3)

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 \((i,j,k,m,n_1,\dots ,n_{q-2})\). 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 \((i,0,k,m,0,\dots ,0)\). Moreover, from (3), \(0\le m\le q-1\);

  • \(G_2\) is the subset of G(P) corresponding to the coefficients \((i,j,k,m,0,\dots ,0)\) such that \(1\le j\le q-1\), \(k\le q-1\) and \(j+m\le q-1\);

  • \(G_3\) is the subset of G(P) corresponding to the coefficients \((i,j,k,0,\dots ,n_s,\dots ,0)\) such that \(1\le s\le q-2\), \(n_s=1\) and \(i+k+(s+1)q\ge q^2-1\).

Result 8

[3, Observation 4.4] For a point \(P\in \mathcal {X}(\overline{\mathbb {F}}_q){\setminus } \mathcal {X}(\mathbb {F}_{q^6})\), \(\max \{m \in \mathbb {Z} \ : \ m-1 \notin H(P)\}=2\mathfrak {g}-q^2+2\).

3 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 \(T\subset H(P)\) and that T and H(P) have the same genus. For this purpose, some properties of the following subsets of T are useful.

$$\begin{aligned} Ap_1:= & {} \{ a(q^3-1)+i(q^3-q)+j(q^4-q^3-q^2) \mid a=2,\dots ,q-1,\\&\quad \,\, i=0,\dots , q-1, \quad j=0,\dots , a-2 \};\\ Ap_{2,1}:= & {} \{q^3+i(q^3-q)+j(q^4-q^3-q^2) \mid \\&\quad i=0,\dots , q-1, \quad j=0,\dots , q-1\};\\ Ap_{2,2}:= & {} \{(q^3-1)+i(q^3-q)+j(q^4-q^2-1)\mid \\&\quad i=0,\dots , q-1, \quad j=0,\dots , q-2 \};\\ Ap_2:= & {} (Ap_{2,1}{\setminus } \{q^3\})\cup Ap_{2,2};\\ Ap_3:= & {} \{q^3+q^3-1+i(q^3-q)+j(q^4-q^3-q^2)\mid \\&\quad i,j=0,\dots , q-1, \quad j\ne 0\}\\ Ap_4:= & {} \{q^3+a(q^3-1)+i(q^3-q)+j(q^4-q^3-q^2) \mid \\&\quad i=0,\dots , q-1,\\&\quad j=2,\dots , q-1 , \quad a=2,\dots ,j \};\\ A:= & {} Ap_1\cup Ap_2\cup Ap_3\cup Ap_4\cup \{0\}. \end{aligned}$$

Proposition 2

The sets \(Ap_1\), \(Ap_{2,1}\), \(Ap_{2,2}\), \(Ap_3\), and \(Ap_4\) are pairwise disjoint.

Proof

Let \(x_{a,i,j}\) denote the element of \(Ap_1\) corresponding to the choices of the parameters a, i, j, that is

$$\begin{aligned} x=a(q^3-1)+i(q^3-q)+j(q^4-q^3-q^2). \end{aligned}$$

We use an analogous notation for the elements of \(Ap_{2,1}\), \(Ap_{2,2}\), \(Ap_3\) and \(Ap_4\).

  • \(Ap_1\cap Ap_{2,1}\) is empty since no element of \(Ap_1\) is divisible by q. The same argument also shows that \(Ap_{2,1}\cap Ap_{2,2}\), \(Ap_{2,1}\cap Ap_{3}\) and \(Ap_{2,1}\cap Ap_{4}\) are empty.

  • Let \(x_{a,i,j} \in Ap_1\) and \(x_{\bar{i},\bar{j}} \in Ap_{2,2}\). If \(x_{a,i,j}=x_{\bar{i},\bar{j}}\) then

    $$\begin{aligned}&a(q^3-1)+i(q^3-q)+j(q^4-q^3-q^2)\nonumber \\&\quad = (q^3-1)+\bar{i}(q^3-q)+\bar{j}(q^4-q^2-1). \end{aligned}$$
    (4)

    Reducing Eq. (4) modulo q we obtain \(a=\bar{j}+1\). Substituting \(a=\bar{j}+1\) in (4) and dividing by q it is readily seen (again reducing modulo q) that \(i=\bar{i}\). Thus Eq. (4) now reads

    $$\begin{aligned} j(q^2-q-1)=\bar{j}(q^2-q-1), \end{aligned}$$

    whence \(j=\bar{j}\), a contradiction since \(j\le a-2=\bar{j}-1\).

  • Let \(x_{a,i,j} \in Ap_1\) and \(x_{\bar{i},\bar{j}} \in Ap_{3}\). If \(x_{a,i,j}=x_{\bar{i},\bar{j}}\) then

    $$\begin{aligned}&a(q^3-1)+i(q^3-q)+j(q^4-q^3-q^2)\\&\quad = 2q^3-1+\bar{i}(q^3-q)+\bar{j}(q^4-q^3-q^2), \end{aligned}$$

    that modulo q yields \(a=1\), a contradiction with \(a\ge 2\).

  • Let \(x_{a,i,j} \in Ap_1\) and \(x_{\bar{a},\bar{i},\bar{j}} \in Ap_{4}\). If \(x_{a,i,j}=x_{\bar{a},\bar{i},\bar{j}}\) then

    $$\begin{aligned}&a(q^3-1)+i(q^3-q)+j(q^4-q^3-q^2)\\&\quad = q^3+\bar{a}(q^3-1)+\bar{i}(q^3-q)+\bar{j}(q^4-q^3-q^2), \end{aligned}$$

    that modulo q yields \(a=\bar{a}\). Therefore

    $$\begin{aligned} i(q^3-q)+j(q^4-q^3-q^2)= q^3+\bar{i}(q^3-q)+\bar{j}(q^4-q^3-q^2), \end{aligned}$$

    whence \(i=\bar{i}\) follows. Thus

    $$\begin{aligned} j(q^4-q^3-q^2)=q^3+\bar{j}(q^4-q^3-q^2), \end{aligned}$$

    whence \(j\ge \bar{j}\), a contradiction with \(j\le a-2=\bar{a}-2\le \bar{j}-2\).

  • \(Ap_{2,2}\cap Ap_{3}\) is empty since for every element x of \(Ap_3\), \(x-(q^3-1)\) is divisible by q, whereas this fails for any element of \(Ap_{2,2}\).

  • Let \(x_{i,j} \in Ap_{2,2}\) and \(x_{\bar{a},\bar{i},\bar{j}} \in Ap_{4}\). If \(x_{i,j}=\bar{x}_{\bar{a},\bar{i},\bar{j}}\) then

    $$\begin{aligned}&(q^3-1)+i(q^3-q)+j(q^4-q^2-1)\nonumber \\&\quad = q^3+\bar{a}(q^3-1)+\bar{i}(q^3-q)+\bar{j}(q^4-q^3-q^2), \end{aligned}$$
    (5)

    whence reducing modulo q yields \(j=\bar{a}-1\). Now Equation (6) reads

    $$\begin{aligned} i(q^2-1)+j(q^3-q^2-q)-q^2= \bar{i}(q^2-1)+\bar{j}(q^3-q^2-q), \end{aligned}$$
    (6)

    and hence \(i=\bar{i}\). Therefore

    $$\begin{aligned} j(q^3-q^2-q)=q^2+\bar{j}(q^3-q^2-q) \end{aligned}$$

    and \(j\ge \bar{j}\), a contradiction with \(j=\bar{a}-1\le \bar{j}-1\).

  • \(Ap_3\cap Ap_{4}\) is empty since for every element x of \(Ap_3\), \(x+1\) is divisible by q, but this fails for any element of \(Ap_4\).

\(\square\)

Proposition 3

The cardinalities of the sets \(Ap_1,Ap_2,Ap_3,Ap_4\) are as follows

  1. (i)

    \(|Ap_1|=|Ap_4|=q(q-1)(q-2)/2\);

  2. (ii)

    \(|Ap_2|=q^2+q(q-1)-1\);

  3. (iii)

    \(|Ap_3|=q(q-1)\);

  4. (iv)

    \(|A|=q^3\).

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 \(x,y \in Ap_1\), so

$$\begin{aligned} x=a(q^3-1)+i(q^3-q)+j(q^4-q^3-q^2) \end{aligned}$$

and

$$\begin{aligned} y=\bar{a}(q^3-1)+\bar{i}(q^3-q)+\bar{j}(q^4-q^3-q^2), \end{aligned}$$

with \(a,\bar{a}\in \{2,\dots ,q-1\}\), \(i,\bar{i}\in \{0,\dots , q-1\}\), and \(j\in \{0,\dots , a-2\}\), \(\bar{j}\in \{0,\dots , \bar{a}-2\}\). Assume that \(x=y\) holds. Then \(a\equiv \bar{a} \pmod {q}\), and since \(a,\bar{a}\in \{2,\dots ,q-1\}\), we obtain \(a=\bar{a}\). Therefore

$$\begin{aligned} i(q^3-q)+j(q^4-q^3-q^2)=\bar{i}(q^3-q)+\bar{j}(q^4-q^3-q^2), \end{aligned}$$

whence

$$\begin{aligned} i(q^2-1)+j(q^3-q^2-q)=\bar{i}(q^2-1)+\bar{j}(q^3-q^2-q). \end{aligned}$$

By applying the same argument as above, we obtain \(i=\bar{i}\). Finally, this implies \(j=\bar{j}\), and so the claim follows. \(\square\)

Proposition 4

If \(x\in A\) then \(x-q^3\not \in H(P)\).

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,

  1. (a)

    if \(x \in Ap_1\) then

    $$\begin{aligned}&x-q^3=a(q^3-1)+i(q^3-q)+j(q^4-q^3-q^2)-q^3\\&\quad =(a+i+jq-j-2)q^3+(q-j-1)q^2+(q-i-1)q+q-a-1 +1, \end{aligned}$$

    where \(a\in \{2,\dots ,q-1\}\), \(i\in \{0,\dots , q-1\}\) and \(j\in \{0,\dots , a-2\}\). Therefore

    $$\begin{aligned} {\left\{ \begin{array}{ll} a+i+jq-j-2\ge 0\\ q-j-1\ge 0 \\ q-i-1\ge 0 \\ 0\le q-a-1\le q-1\\ (a+i+jq-j-2)+q(q-j-1)-(q-j-2)+(q-i-1)+\\ +(q-a-1)=q^2-2.\\ \end{array}\right. } \end{aligned}$$

    Therefore \(x-q^3\not \in H(P)\) by (3).

  2. (b)

    if \(x \in Ap_{2,1}{\setminus } \{q^3\}\) then

    $$\begin{aligned} x-q^3= & {} i(q^3-q)+j(q^4-q^3-q^2)\nonumber \\= & {} (i+jq-j-1)q^3+(q-j-1)(q^2+1)+(q-i-1)q+j+1; \end{aligned}$$
    (7)

    where \(i\in \{0,\dots , q-1\}\) and \(j\in \{0,\dots , q-1\}\). Since \(x\ne q^3\), \((i,j)\ne (0,0)\) and

    $$\begin{aligned} {\left\{ \begin{array}{ll} i+jq-j-1\ge 0\\ q-j-1\ge 0 \\ q-i-1\ge 0\\ 0\le j\le q-1\\ i+jq-j-1+q(q-j-1)+(q-i-1)+j=q^2-2.\\ \end{array}\right. } \end{aligned}$$

    Therefore \(x-q^3\not \in H(P)\) by (3).

  3. (c)

    if \(x \in Ap_{2,2}\) then

    $$\begin{aligned} x-q^3= & {} i(q^3-q)+j(q^4-q^2-1)-1\nonumber \\= & {} (i+jq-1)q^3+(q-j-2)(q^2+1)+(2q-i-1)q+1; \end{aligned}$$
    (8)

    where \(i\in \{0,\dots , q-1\}\) and \(j\in \{0,\dots , q-2\}\). Now if \((i,j)=(0,0)\) then \(x=q^3-1\) and hence \(x-q^3\not \in H(P)\). Therefore \((i,j)\ne (0,0)\) is assumed. Then

    $$\begin{aligned} {\left\{ \begin{array}{ll} i+jq-1\ge 0 \\ q-j-2\ge 0\\ 2q-i-1\ge 0\\ i+jq-1+q(q-j-2)+(2q-i-1)=q^2-2.\\ \end{array}\right. } \end{aligned}$$

    Therefore \(x-q^3\not \in H(P)\) by (3).

  4. (d)

    if \(x \in Ap_3\) then

    $$\begin{aligned}&x-q^3= q^3-1+i(q^3-q)+j(q^4-q^3-q^2)\\&\quad =(i+jq-j)q^3+(q-j-1)(q^2+1)+(q-i-1)q+ j-1+1; \end{aligned}$$

    where \(i\in \{0,\dots , q-1\}\) and \(j\in \{1,\dots , q-1\}\). Therefore

    $$\begin{aligned} {\left\{ \begin{array}{ll} i+jq-j\ge 0\\ q-j-1\ge 0\\ q-i-1\ge 0\\ 0\le j-1 \le q-1\\ i+jq-j+q(q-j-1)+(q-i-1)+j-1=q^2-2.\\ \end{array}\right. } \end{aligned}$$

    Therefore \(x-q^3\not \in H(P)\) by (3).

  5. (e)

    if \(x \in Ap_4\) then

    $$\begin{aligned}&x-q^3= a(q^3-1)+i(q^3-q)+j(q^4-q^3-q^2)\\&\quad =(i+jq-j+a-1)q^3\\&\quad \quad +(q-j-1)(q^2+1)+(q-i-1)q+ j-a+1; \end{aligned}$$

    where \(i\in \{0,\dots , q-1\}\), \(j\in \{2,\dots , q-1\}\) and \(a\in \{2,\dots ,j\}\). Therefore

    $$\begin{aligned} {\left\{ \begin{array}{ll} i+jq-j+a-1\ge 0\\ q-j-1\ge 0\\ q-i-1\ge 0\\ 0\le j-a\le q-1\\ i+jq-j+a-1+q(q-j-1)+(q-i-1)+j-a=q^2-2.\\ \end{array}\right. } \end{aligned}$$

    Therefore \(x-q^3\not \in H(P)\) by (3).

\(\square\)

We use Proposition 4 to prove the following lemma.

Lemma 1

The semigroup T is contained in H(P).

Proof

Since \(T=\langle S\rangle\), it suffices to show that \(S=S_1 \cup S_2\subseteq H(P)\). We carry out the computation for the case \(x\in S_1=Ap_{2,1}\). Analogous computation can be done for the other elements in \(S_2=Ap_{2,2}\). Take \(x\in S_1\). Then

$$\begin{aligned} x=q^3+i(q^3-q)+j(q^4-q^3-q^2), \end{aligned}$$

for some \(0\le i\le q-1\) and \(0\le j \le q-1\). It may be observed that

$$\begin{aligned} x=(i+jq-j)q^3+(q-j-1)(q^2+1)+(q-i-1)q+j+1. \end{aligned}$$

We assume on the contrary \(x\in G(P)\). Taking into account Result 7 we distinguish three cases according to either \(x\in G_1\), or \(x \in G_2\), or \(x \in G_3\).

  • Case \(x\in G_1\). There exist non-negative integers \(\bar{m},\bar{i},\bar{k}\) such that \(\bar{i}+\bar{k}+\bar{m}q\le q^2-2\) and

    $$\begin{aligned}&(i+jq-j)q^3+(q-j-1)(q^2+1)+(q-i-1)q+j+1\nonumber \\&\quad =\bar{i}q^3+\bar{m}(q^2+1)+\bar{k}q+1. \end{aligned}$$
    (9)

    Equation (9) modulo q yields

    $$\begin{aligned} \bar{m}\equiv -1 \pmod q, \end{aligned}$$

    whence \(\bar{m}=q-1\). Hence

    $$\begin{aligned} (i+jq-j)q^3+(q-j-1)q^2+(q-i-1)q=\bar{i}q^3+\bar{m}q^2+\bar{k}q, \end{aligned}$$

    and, dividing by q,

    $$\begin{aligned} (i+jq-j)q^2+(q-j-1)q+q-i-1=\bar{i}q^2+(q-1)q+\bar{k}, \end{aligned}$$

    that is

    $$\begin{aligned} (i+jq-j)q^2-jq+q-i-1=\bar{i}q^2+\bar{k}. \end{aligned}$$
    (10)

    Equation (10) modulo q now yields

    $$\begin{aligned} \bar{k} \equiv -i-1 \pmod q. \end{aligned}$$

    Moreover \(\bar{i}+\bar{k}+\bar{m}q\le q^2-2\), gives \(\bar{k}+\bar{i}\le q-2\) and hence \(\bar{k}=q-i-1\).

    Substituting in Eq. (10) we obtain

    $$\begin{aligned} (i+jq-j)q^2-jq=\bar{i}q^2. \end{aligned}$$

    Again dividing by q and reducing shows \(j\equiv 0 \pmod q\), whence \(j=0\). Therefore \(\bar{i}=i\), and a contradiction arises from \(\bar{k}+\bar{i}\le q-2\).

  • Case \(x\in G_2\). There exist non-negative integers \(\bar{m},\bar{i},\bar{k}\) and \(\bar{j}\) such that

    $$\begin{aligned} {\left\{ \begin{array}{ll} 1\le \bar{j}\le q-1,\\ \bar{k}\le q-1,\\ \bar{j}+\bar{m}\le q-1,\\ \bar{i}+\bar{k}+\bar{j}+\bar{m}q\le q^2-2\\ \end{array}\right. } \end{aligned}$$

    and

    $$\begin{aligned}&(i+jq-j)q^3+(q-j-1)(q^2+1)+(q-i-1)q+j+1\nonumber \\&\quad =\bar{i}q^3+\bar{m}(q^2+1)+\bar{k}q+\bar{j}+1. \end{aligned}$$
    (11)

    Then, reducing modulo q, Eq. (11) yields \(\bar{j}+\bar{m}\equiv -1 \pmod q\). As \(\bar{j}+\bar{m}\le q-1\), we have \(\bar{j}+\bar{m}= q-1\) and (11) reads

    $$\begin{aligned}&(i+jq-j)q^3+(q-j-1)(q^2+1)+(q-i-1)q+j+1\\&\quad =\bar{i}q^3+\bar{m}q^2+\bar{k}q+\bar{m}+\bar{j}+1, \end{aligned}$$

    that is

    $$\begin{aligned} (i+jq-j)q^2+(q-j-1)q+q-i-1=\bar{i}q^2+\bar{m}q+\bar{k}. \end{aligned}$$
    (12)

    Again, \(\bar{k}\le q-1\) and Eq. (12) modulo q imply \(\bar{k}=q-i-1\). Thus

    $$\begin{aligned} (i+jq-j)q+(q-j-1)=\bar{i}q+\bar{m}, \end{aligned}$$
    (13)

    whence \(\bar{m}=q-j-1\) and \(\bar{j}=j\). Finally, \(\bar{i}=i+jq-j\) and

    $$\begin{aligned}&\bar{i}+\bar{k}+\bar{j}+\bar{m}q=i+jq-j+q-i-1\\&\quad + j+(q-j-1)q = q^2-1>q^2-2, \end{aligned}$$

    a contradiction.

  • Case \(x\in G_3\). There exist non-negative integers \(s,\bar{i},\bar{k}\) and \(\bar{j}\) such that

    $$\begin{aligned} {\left\{ \begin{array}{ll} 1\le s\le q-2,\\ \bar{j},\bar{k}\le q-1,\\ \bar{i}+\bar{k}+(s+1)q\ge q^2-1,\\ \bar{i}+\bar{j}+\bar{k}+(s+1)q-s\le q^2-2\\ \end{array}\right. } \end{aligned}$$

    and

    $$\begin{aligned}&(i+jq-j)q^3+(q-j-1)(q^2+1)+(q-i-1)q+j+1\nonumber \\&\quad =\bar{i}q^3+(s+1)q^2+\bar{k}q+\bar{j}+1. \end{aligned}$$
    (14)

    Note that in particular \(\bar{j}<s\) must hold. On the other hand, Eq. (14) modulo q yields \(\bar{j}=q-1>s\), a contradiction.

\(\square\)

Proposition 5

\(A=Ap(H(P),q^3)=Ap(T,q^3)\).

Proof

It is readily seen that each element of A is a linear combination of elements of S. Therefore \(A\subset T\) and by Propositions 3 and 4 we get \(A=Ap(H(P),q^3)\). Moreover, from Lemma 1 we have \(T\subseteq H(P)\) so each gap of H(P) is also a gap T, whence the claim follows. \(\square\)

Now Result 3 and Proposition 5 show that T and H(P) have the same genus. Furthermore, since T is contained in H(P), \(T=\langle S \rangle = H(P)\). Finally, since \(S=Ap_2\cup \{q^3\}\), Proposition 3 yields \(|S|=e(H(P))=2q^2-q\). This ends the proof of Theorem 1.

4 AG codes from \(\mathbb {F}_{q^7}\)-rational points of the GK curve

In this section we construct a family of AG codes from \(\mathbb {F}_{q^7}\)-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 \(P\in {\mathcal {X}}(\mathbb {F}_{q^7}){\setminus }{\mathcal {X}}(\mathbb {F}_{q})\), \(H(P)=\lbrace 0=\rho _1<\rho _2<...\rbrace\) denotes the Weierstrass semigroup at P and \(C_{\ell }(P)\) stands for the dual code \(C_{\ell }(P) =C^{\perp }(D,\rho _{\ell }P)\), where

$$\begin{aligned} D=\sum _{Q\in {\mathcal {X}}(\mathbb {F}_{q^7}) {\setminus }{\{P\}} } Q \end{aligned}$$

is a divisor supported at all \(\mathbb {F}_{q^7}\)-rational points of \({\mathcal {X}}\) but P. From the Feng–Rao lower bound on the minimum distance of \(C_{\ell }(P)\), we have that \(C_{\ell }(P)\) is an \([n,k,d]_{q^7}\) linear code, with \(n=q^7\), \(k=n-\ell\) and

$$\begin{aligned} d\ge \max \{d_{ORD}(C_{\ell }(P)),d^*\}, \end{aligned}$$
(15)

where \(d^*=\deg (G)-2\mathfrak {g}+2\) denotes the designed minimum distance of \(C_{\ell }(P)\). 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.

Proposition 6

For every \(\ell \ge 3\mathfrak {g}-2q^2+3\), \(d_{ORD}(C_{\ell }(P))=\ell +1-\mathfrak {g}\).

Remark 1

Proposition 6 also shows that if \(\ell \ge 3\mathfrak {g}-2q^2+3\), then \(d_{ORD}(C_{\ell }(P))=d^*\). Indeed, let \(\ell = 3\mathfrak {g}-2q^2+3+r\) for some \(r\ge 0\). Then \(\ell = \mathfrak {g}+1+(2\mathfrak {g}-2q^2+2+r)\ge \mathfrak {g}+1\). Since \(\rho _{\mathfrak {g}+1}=2\mathfrak {g}\) and Result 8 yields that \(2\mathfrak {g}-q^2+1\) is the largest gap in H(P), we have

$$\begin{aligned} \rho _{\ell }=2\mathfrak {g}+(2\mathfrak {g}-2q^2+r+2)=4\mathfrak {g}-2q^2+r+2. \end{aligned}$$

Hence Proposition 6 yields

$$\begin{aligned} d_{ORD}(C_{\ell }(P))=\ell +1-\mathfrak {g}=2\mathfrak {g}-2q^2+4+r=\rho _\ell -2\mathfrak {g}+2=d^*. \end{aligned}$$

In the remaining cases \(\ell < 3\mathfrak {g}-2q^2+3\) and the Feng–Rao minimum distance may provide an improvement on the designed minimum distance \(d^*\).

For \(q=3\) the parameters of the codes \(C_{\ell }(P)\) are reported in the table below. These codes have length \(n=2187\), whereas their dimension k and their Feng–Rao minimum distance \(d_{ORD}\) varies. We limit ourselves to the cases where \(d_{ORD}(C_{\ell }(P))>d^*\) and by Remark 1 this can only happen when \(\ell < 3\mathfrak {g}-2q^2+3\). As the table shows, the Feng–Rao minimum distance is strictly greater than the designed minimum distance \(d^*\), for all those cases apart from a small number of exceptions.

n

k

\(\rho _{\ell }\)

\(d_{ORD}\)

k

\(\rho _{\ell }\)

\(d_{ORD}\)

k

\(\rho _{\ell }\)

\(d_{ORD}\)

2187

2185

26

2

2184

27

2

2183

50

2

2187

2182

51

2

2181

52

2

2180

53

2

2187

2179

54

2

2178

72

2

2177

74

2

2187

2176

75

2

2175

76

2

2174

77

2

2187

2173

78

2

2172

79

2

2171

80

2

2187

2170

81

2

2169

96

2

2168

97

2

2187

2167

98

2

2166

99

2

2165

100

2

2187

2164

101

2

2163

102

2

2162

103

2

2187

2161

104

2

2160

105

2

2159

106

2

2187

2158

107

2

2157

108

2

2156

117

2

2187

2155

120

2

2154

121

2

2153

122

2

2187

2152

123

2

2151

124

2

2150

125

2

2187

2149

126

2

2148

127

2

2147

128

2

2187

2146

129

2

2145

130

2

2144

131

2

2187

2143

132

2

2142

133

2

2141

134

2

2187

2140

135

2

2139

141

2

2138

143

2

2187

2137

144

2

2136

145

2

2135

146

2

2187

2134

147

2

2133

148

2

2132

149

2

2187

2131

150

2

2130

151

2

2129

152

2

2187

2128

153

2

2127

154

2

2126

155

2

2187

2125

156

2

2124

157

2

2123

158

2

n

k

\(\rho _{\ell }\)

\(d_{ORD}\)

k

\(\rho _{\ell }\)

\(d_{ORD}\)

k

\(\rho _{\ell }\)

\(d_{ORD}\)

2187

2122

159

2

2121

160

2

2120

161

2

2187

2119

162

2

2118

165

6

2117

167

8

2187

2116

168

8

2115

169

8

2114

170

8

2187

2113

171

8

2112

172

8

2111

173

8

2187

2110

174

8

2109

175

8

2108

176

8

2187

2107

177

8

2106

178

8

2105

179

8

2187

2104

180

8

2103

181

8

2102

182

8

2187

2101

183

8

2100

184

8

2099

185

8

2187

2098

186

8

2097

187

8

2096

188

8

2187

2095

189

8

2094

191

11

2093

192

14

2187

2092

193

19

2091

194

19

2090

195

19

2187

2089

196

19

2088

197

19

2087

198

19

2187

2086

199

19

2085

200

19

2084

201

19

2187

2083

202

19

2082

203

19

2081

204

19

2187

2080

205

19

2079

206

19

2078

207

19

2187

2077

208

19

2076

209

19

2075

210

19

2187

2074

211

19

2073

212

19

2072

213

19

2187

2071

214

19

2068

217

28

2067

218

34

2187

2066

219

38

2065

220

43

2064

221

43

2187

2063

222

43

2062

223

43

2061

224

43

2187

2060

225

43

2059

226

43

2058

227

43

2187

2057

228

43

2056

229

43

2055

230

43

2187

2054

231

43

2053

232

43

2052

233

43

2187

2051

234

43

2050

235

43

2049

236

43

2187

2048

237

43

2047

238

43

2041

244

54

n

k

\(\rho _{\ell }\)

\(d_{ORD}\)

k

\(\rho _{\ell }\)

\(d_{ORD}\)

k

\(\rho _{\ell }\)

\(d_{ORD}\)

2187

2040

245

59

2039

246

62

2038

247

65

2187

2037

248

65

2036

249

65

2035

250

65

2187

2034

251

65

2033

252

65

2032

253

65

2187

2031

254

65

2030

255

65

2029

256

65

2187

2028

257

65

2027

258

65

2026

259

65

2187

2025

260

65

2023

262

67

2014

271

80

2187

2013

272

84

2012

273

86

2011

274

90

2187

2010

275

92

2009

276

92

2008

277

92

2187

2007

278

92

2006

279

92

2005

280

92

We point out that many other linear codes can be obtained from the above table by using the following propagation rules; see [23, Exercise 7].

Result 9

If an \([n,k,d]_q\) linear code exists, then:

  1. (i)

    for every non-negative integer \(s<d\), an \([n,k,d-s]_q\) linear code exists;

  2. (ii)

    for every non-negative integer \(s<k\), an \([n,k-s,d]_q\) linear code exists;

  3. (iii)

    for every non-negative integer \(s<k\), an \([n-s,k-s,d]_q\) linear code exists;

  4. (iv)

    for every non-negative integer \(s<\min \{n-k-1,d\}\), an \([n-s,k,d-s]_q\) linear code exists.

5 Quantum codes from \(\mathbb {F}_{q^7}\)-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 \(\mathbb {H}=(\mathbb {C}^q)^{\otimes n}=\mathbb {C}^q \otimes \cdots \otimes \mathbb {C}^q\) 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 \(\mathbb {H}\). 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 \(\lfloor \frac{d-1}{2}\rfloor\) errors. Moreover, the quantum version of the Singleton bound states that for a \([[n,k,d]]_q\)-quantum code, \(2d+k\le 2+n\) holds. Again, by analogy with the ordinary case, the quantum Singleton defect and the relative quantum Singleton defect are defined to be \(\delta _Q:= n-k-2d+2\) and \(\varDelta _Q:=\frac{\delta _Q}{n}\), respectively. We recall [14, Lemma 2.5].

Lemma 2

(CSS construction) Let \(C_1\) and \(C_2\) be linear codes with parameters \([n,k_1,d_1]_q\) and \([n,k_2,d_2]_q\), respectively, and assume that \(C_1 \subset C_2\). Then there exists a \([[n,k_2-k_1,d]]_q\)-quantum code with

$$\begin{aligned} d=\min \{ w(c)\, \vert \, c\in (C_2 {\setminus } C_1)\cup (C_1^{\perp } {\setminus } C_2^{\perp }) \}. \end{aligned}$$

We apply the CSS construction to the dual codes \(C_\ell (P)\) constructed in Sect. 4. We keep the same notation as in Sect. 4. For two non-gaps \(\rho _\ell , \rho _{\ell +s}\in H(P)\), with \(s\ge 1\), let \(C_1=C_{\ell +s}(P)\) and \(C_2=C_\ell (P)\) be the codes constructed in Sect. 4. Then \(C_1\subset C_2\). Also, if \(k_i\) denotes the dimension of \(C_i\), then

$$\begin{aligned} k_2=q^7-h_{\ell } \quad \text { and } \quad k_1=q^7 -h_{\ell +s}=q^7-h_\ell -s, \end{aligned}$$

where \(h_i\) is the number of those non-gaps at P that do not exceed i. The CSS construction now provides a \([[n,s,d]]_{q^7}\)-quantum code with \(n=q^7\) and

$$\begin{aligned} d =\min \lbrace w(c)\, \vert \, c \in (C_\ell {\setminus } C_{\ell +s})\cup (C(D,\rho _{\ell +s}P){\setminus } C(D,\rho _\ell P)) \rbrace . \end{aligned}$$

It may be noted that

$$\begin{aligned} d\ge \min \lbrace d_{ORD}(C_\ell ),d_1 \rbrace , \end{aligned}$$
(16)

where \(d_1\) is the minimum distance of \(C(D,\rho _{\ell +s}P)\).

Theorem 10

For every \(\ell \in [3\mathfrak {g}-2q^2+3,q^7-\mathfrak {g}]\) and \(s\in [1,q^7-2\ell ]\) there exists a \([[q^7,s,d]]_{q^7}\)-quantum code with \(d\ge \ell +1-\mathfrak {g}\).

Proof

Since \(\ell \ge 3\mathfrak {g}-2q^2+3\), Proposition 6 applies and \(d_{ORD}(C_\ell )=\ell +1-\mathfrak {g}\). Also, \(\rho _{\ell +s}=\mathfrak {g}-1+\ell +s\), whence \(d_1\ge q^7-\deg (\rho _{l+s}P)=q^7-\rho _{\ell +s}\ge q^7-\ell -s-\mathfrak {g}+1\). Since \(s\le q^7-2\ell\), then \(d_{ORD}(C_\ell )\le d_1\) and the claim follows from (16). \(\square\)

For \(\ell \in [3\mathfrak {g}-2q^2+3,q^7-\mathfrak {g}]\) and \(s=q^7-2\ell\), Theorem 10 proves the existence of \([[q^7,s,d]]_{q^7}\)-quantum codes whose relative quantum Singleton defect \(\varDelta _Q\) is upper bounded as follows,

$$\begin{aligned} \varDelta _Q=\frac{q^7-s-2d+2}{q^7}=\frac{2\ell -2d+2}{q^7}\le \frac{2\mathfrak {g}}{q^7}=\frac{q^5-2q^3+q^2}{q^7}, \end{aligned}$$

and therefore it goes to 0 as q goes to infinity.

For \(q=3\) and \(\ell\) ranging in \(\mathfrak {g},\ldots ,3\mathfrak {g}-2q^2+2\) the following table reports the parameters of quantum codes which are the first non-trivial cases in which Theorem 10 does not apply.

n

s

\(d\ge\)

s

\(d\ge\)

s

\(d\ge\)

s

\(d\ge\)

2187

1989

1

1987

2

1985

3

1983

4

2187

1981

5

1979

6

1977

7

1975

8

2187

1973

9

1971

10

1969

11

1967

12

2187

1965

13

1963

14

1961

15

1959

16

2187

1957

17

1955

18

1953

19

1951

20

2187

1949

21

1947

22

1945

23

1943

24

2187

1941

25

1939

26

1937

27

1935

28

2187

1933

29

1931

30

1929

31

1927

32

2187

1925

33

1923

34

1921

35

1919

36

2187

1917

37

1915

38

1913

39

1911

40

2187

1909

41

1907

42

1905

43

1903

44

2187

1901

45

1899

46

1897

47

1895

48

2187

1893

49

1891

50

1889

51

1887

52

2187

1885

53

1883

54

1881

55

1879

56

2187

1877

57

1875

58

1873

59

1871

60

2187

1869

61

1867

62

1865

63

1863

64

2187

1861

65

1859

66

1857

67

1855

68

2187

1853

69

1851

70

1849

71

1847

72

2187

1845

73

1843

74

1841

75

1839

76

2187

1837

77

1835

78

1833

79

1831

80

2187

1829

81

1827

82

1825

83

1823

84

n

s

\(d\ge\)

s

\(d\ge\)

s

\(d\ge\)

s

\(d\ge\)

2187

1821

85

1819

86

1817

87

1815

88

2187

1813

89

1811

90

1809

91

1807

92

2187

1805

93

1803

94

1801

95

1799

96

2187

1797

97

1795

98

1793

99

1791

100

2187

1789

101

1787

102

1785

103

1783

104

2187

1781

105

1779

106

1777

107

1775

108

2187

1773

109

1771

110

1769

111

1767

112

2187

1765

113

1763

114

1761

115

1759

116

2187

1757

117

1755

118

1753

119

1751

120

2187

1749

121

1747

122

1745

123

1743

124

2187

1741

125

1739

126

1737

127

1735

128

2187

1733

129

1731

130

1729

131

1727

132

2187

1725

133

1723

134

1721

135

1719

136

2187

1717

137

1715

138

1713

139

1711

140

2187

1709

141

1707

142

1705

143

1703

144

2187

1701

145

1699

146

1697

147

1695

148

2187

1693

149

1691

150

1689

151

1687

152

2187

1685

153

1683

154

1681

155

1679

156

2187

1677

157

1675

158

1673

159

1671

160

2187

1669

161

1667

162

1665

163

1663

164

2187

1661

165

1659

166

1657

167

1655

168

2187

1653

169

1651

170

1649

171

1647

172

2187

1645

173

1643

174

1641

175

1639

176

2187

1637

177

1635

178

1633

179

1631

180

2187

1629

181

1627

182

1625

183

  

We end this section with the construction of a second family of quantum codes arising from the GK curve. Our construction is based on a generalization of Lemma 2 given in [14, Theorem 3.1].

Lemma 3

(General t-point construction) Let \(\mathcal {Y}\) be an absolutely irreducible non-singular curve over \(\mathbb {F}_q\) of genus \(\mathfrak {g}\) containing \(n+t\) distinct \(\mathbb {F}_q\)-rational points for some \(n,t>0\). For every \(i=1,\ldots ,t\), let \(a_i,b_i\) be positive integers such that \(a_i\le b_i\) and that

$$\begin{aligned} 2\mathfrak {g}-2<\sum _{i=1}^t a_i<\sum _{i=1}^t b_i < n. \end{aligned}$$

Then there exists a \([[n,k,d]]_q\)-quantum code with \(k=\sum _{i=1}^t b_i-\sum _{i=1}^t a_i\) and \(d\ge \min \lbrace n-\sum _{i=1}^t b_i, \sum _{i=1}^t a_i-(2\mathfrak {g}-2)\rbrace\).

Lemma 3 applied to the set of \(\mathbb {F}_{q^7}\)-rational points of the GK curve gives the following result.

Proposition 7

Let \(a,b\in \mathbb {N}_0\) such that

$$\begin{aligned} q^5-2q^3+q^2-2<a<b<q^7. \end{aligned}$$

Then there exists a quantum code with parameters \([[q^7,b-a,d]]_{q^7}\) , where

$$\begin{aligned} d\ge \min \lbrace q^7-b,\; a-(q^5-2q^3+q^2-2) \rbrace . \end{aligned}$$