Skip to main content
Top
Published in: Applicable Algebra in Engineering, Communication and Computing 4/2023

Open Access 04-09-2021 | Original Paper

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

Authors: Stefano Lia, Marco Timpanella

Published in: Applicable Algebra in Engineering, Communication and Computing | Issue 4/2023

Activate our intelligent search to find suitable subject content or patents.

search-config
loading …

Abstract

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 \({\mathcal {X}}\) were investigated and the sets of minimal generators were determined for all points in \({\mathcal {X}}(\mathbb {F}_{q^2})\) and \({\mathcal {X}}(\mathbb {F}_{q^6})\setminus {\mathcal {X}}( \mathbb {F}_{q^2})\). This paper completes their work by settling the remaining cases, that is, for points in \({\mathcal {X}}(\overline{\mathbb {F}}_{q}){\setminus }{\mathcal {X}}( \mathbb {F}_{q^6})\). As an application to AG codes, we determine the dimensions and the lengths of duals of one-point codes from a point in \({\mathcal {X}}(\mathbb {F}_{q^7}){\setminus }{\mathcal {X}}( \mathbb {F}_{q})\) and we give a bound on the Feng–Rao minimum distance \(d_{ORD}\). For \(q=3\) we provide a table that also reports the exact values of \(d_{ORD}\). As a further application we construct quantum codes from \(\mathbb {F}_{q^7}\)-rational points of the GK-curve.
Notes

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

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(DG) 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(DG) has length N. Moreover, if \(N>\deg (G)\) then ev is an embedding and \(\ell (G)\) equals the dimension of C(DG). The minimum distance d of C(DG), 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(DG). 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 (DG) 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.
if \({\mathcal {X}}\) is \(\mathbb {F}_{q}\)-maximal and n is odd, then \({\mathcal {X}}\) is \(\mathbb {F}_{q^{n}}\)-maximal;
 
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 aij, 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
(i)
\(|Ap_1|=|Ap_4|=q(q-1)(q-2)/2\);
 
(ii)
\(|Ap_2|=q^2+q(q-1)-1\);
 
(iii)
\(|Ap_3|=q(q-1)\);
 
(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,
(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).
 
(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).
 
(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).
 
(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).
 
(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:
(i)
for every non-negative integer \(s<d\), an \([n,k,d-s]_q\) linear code exists;
 
(ii)
for every non-negative integer \(s<k\), an \([n,k-s,d]_q\) linear code exists;
 
(iii)
for every non-negative integer \(s<k\), an \([n-s,k-s,d]_q\) linear code exists;
 
(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
$$\begin{aligned} d\ge \min \lbrace q^7-b,\; a-(q^5-2q^3+q^2-2) \rbrace . \end{aligned}$$

Acknowledgements

The research of S. Lia and M. Timpanella was partially supported by the Italian National Group for Algebraic and Geometric Structures and their Applications (GNSAGA - INdAM). The second author was supported by the project “VALERE: VAnviteLli pEr la RicErca” of the University of Campania “Luigi Vanvitelli”.
Open AccessThis 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://​creativecommons.​org/​licenses/​by/​4.​0/​.

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Literature
2.
3.
4.
5.
go back to reference Duursma, I., Kirov, R.: An Extension of the Order Bound for AG Codes. Lecture Notes in Computer Science, vol. 5527. Springer, Berlin (2009)MATH Duursma, I., Kirov, R.: An Extension of the Order Bound for AG Codes. Lecture Notes in Computer Science, vol. 5527. Springer, Berlin (2009)MATH
6.
go back to reference Eid, A., Hasson, H., Ksir, A., Peachey, J.: Suzuki-invariant codes from the Suzuki curve. Des. Codes Cryptogr. 81, 413–425 (2016)MathSciNetCrossRefMATH Eid, A., Hasson, H., Ksir, A., Peachey, J.: Suzuki-invariant codes from the Suzuki curve. Des. Codes Cryptogr. 81, 413–425 (2016)MathSciNetCrossRefMATH
8.
12.
go back to reference Høholdt, T., van Lint, J.H., Pellikaan, R.: Algebraic geometry codes. In: Pless, V.S., Huffman, W.C., Brualdi, R.A. (eds.) Handbook of Coding Theory, vol. 1, pp. 871–961. Elsevier, Amsterdam (1998) Høholdt, T., van Lint, J.H., Pellikaan, R.: Algebraic geometry codes. In: Pless, V.S., Huffman, W.C., Brualdi, R.A. (eds.) Handbook of Coding Theory, vol. 1, pp. 871–961. Elsevier, Amsterdam (1998)
13.
go back to reference Korchmáros, G., Nagy, G.P., Timpanella, M.: Codes and gap sequences of Hermitian curves. IEEE Trans. Inf. Theory (2019) Korchmáros, G., Nagy, G.P., Timpanella, M.: Codes and gap sequences of Hermitian curves. IEEE Trans. Inf. Theory (2019)
14.
go back to reference La Guardia, G.G., Pereira, F.R.F.: Good and asymptotically good quantum codes derived from algebraic geometry codes. Quantum Inf. Process. 16(6), Article ID 165, 12 pages (2017) La Guardia, G.G., Pereira, F.R.F.: Good and asymptotically good quantum codes derived from algebraic geometry codes. Quantum Inf. Process. 16(6), Article ID 165, 12 pages (2017)
15.
16.
go back to reference McGuire, G., Yılmaz, E.S.: Divisibility of L-polynomials for a family of Artin–Schreier curves. J. Pure Appl. Algebra 223, 3341–3358 (2019)MathSciNetCrossRefMATH McGuire, G., Yılmaz, E.S.: Divisibility of L-polynomials for a family of Artin–Schreier curves. J. Pure Appl. Algebra 223, 3341–3358 (2019)MathSciNetCrossRefMATH
17.
18.
go back to reference Montanucci, M., Timpanella, M., Zini, G.: AG codes and AG quantum codes from cyclic extensions of the Suzuki and Ree curves. J. Geom. 109, 23 (2018)MathSciNetCrossRefMATH Montanucci, M., Timpanella, M., Zini, G.: AG codes and AG quantum codes from cyclic extensions of the Suzuki and Ree curves. J. Geom. 109, 23 (2018)MathSciNetCrossRefMATH
19.
go back to reference Pellikaan, R., Shen, B.Z., van Wee, G.J.M.: Which linear codes are algebraic-geometry. IEEE Trans. Inf. Theory 37, 583–602 (1991)CrossRefMATH Pellikaan, R., Shen, B.Z., van Wee, G.J.M.: Which linear codes are algebraic-geometry. IEEE Trans. Inf. Theory 37, 583–602 (1991)CrossRefMATH
20.
go back to reference Rosales, J.C., García-Sánchez, P.A.: Numerical Semigroups, Developments in Mathematics, vol. 20. Springer, New York (2009)CrossRef Rosales, J.C., García-Sánchez, P.A.: Numerical Semigroups, Developments in Mathematics, vol. 20. Springer, New York (2009)CrossRef
21.
go back to reference Sakata, S.: Fast erasure-and-error decoding of any one-point AG codes up to the Feng–Rao bound. Bull. Univ. Electro-Commun. 9, 39–57 (1996)MathSciNet Sakata, S.: Fast erasure-and-error decoding of any one-point AG codes up to the Feng–Rao bound. Bull. Univ. Electro-Commun. 9, 39–57 (1996)MathSciNet
22.
go back to reference Stichtenoth, H.: Algebraic Function Fields and Codes. Springer Stichtenoth, H.: Algebraic Function Fields and Codes. Springer
23.
24.
go back to reference Xing, C., Chen, H.: Improvements on parameters of one-point AG codes from Hermitian curves. IEEE Trans. Inf. Theory 48, 535–537 (2002)MathSciNetCrossRefMATH Xing, C., Chen, H.: Improvements on parameters of one-point AG codes from Hermitian curves. IEEE Trans. Inf. Theory 48, 535–537 (2002)MathSciNetCrossRefMATH
Metadata
Title
AG codes from -rational points of the GK maximal curve
Authors
Stefano Lia
Marco Timpanella
Publication date
04-09-2021
Publisher
Springer Berlin Heidelberg
Published in
Applicable Algebra in Engineering, Communication and Computing / Issue 4/2023
Print ISSN: 0938-1279
Electronic ISSN: 1432-0622
DOI
https://doi.org/10.1007/s00200-021-00519-2

Other articles of this Issue 4/2023

Applicable Algebra in Engineering, Communication and Computing 4/2023 Go to the issue

Premium Partner