AG codes from -rational points of the GK maximal curve
- Open Access
- 04.09.2021
- Original Paper
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Abstract
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].
Anzeige
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 \(S=S_1\cup S_2\), withThen, our main result is the following theorem.
$$\begin{aligned} \begin{array}{llll} S_1 &{}=&{}\{q^3+i(q^3-q)+j(q^4-q^3-q^2)\,|\, i=0,\ldots ,q-1, \quad j=0,\ldots ,q-1 \},\\ S_2 &{}=&{} \{q^3-1+i(q^3-q)+j(q^4-q^2-1)\,|\, i=0,\ldots ,q-1, \quad j=0,\ldots ,q-2 \}. \end{array} \end{aligned}$$
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 \(\{7, 8, 12, 13, 18\}\).
Anzeige
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 \(P\in {\mathcal {X}}\) isThe Riemann-Roch space associated with an \({\mathbb {F}}_q\)-rational divisor D isand its vector space dimension over \(\mathbb {F}_q\) is \(\ell (D)\).
$$\begin{aligned} H(P) := \{n \in \mathbb {N}_0 \ | \ \exists f \in \mathbb {F}_q(\mathcal {X}), (f)_{\infty }=nP\}= \{\rho _0=0<\rho _1<\rho _2<\cdots \}. \end{aligned}$$
$$\begin{aligned} \mathcal {L}(D) := \{ f \in \mathcal {X}(\mathbb {F}_q) \ : \ (f)+D \ge 0\}\cup \{0\} \end{aligned}$$
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,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 \(d^{\bot }\ge \deg {(G)}-2\mathfrak {g}+2\).
$$\begin{aligned} \varOmega (D):=\{\omega \in \varOmega ({\mathcal {X}})\ :\ (\omega )\ge D\}\cup \{0\} \end{aligned}$$
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 isConsiderwith \(P,P_1,\ldots ,P_N\) pairwise distint points in \(\mathcal {X}(\mathbb {F}_q)\). The numberis 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].
$$\begin{aligned} \nu _\ell := | \{(i,j) \in \mathbb {N}_0^2 \ : \ \rho _i+\rho _j = \rho _{\ell +1}\}|. \end{aligned}$$
$$\begin{aligned} {C}_{\ell }(P)= {C}^{\bot }(P_1+P_2+\cdots +P_N,\rho _{\ell }P), \end{aligned}$$
$$\begin{aligned} d_{ORD} ({C}_{\ell }(P)) := \min \{\nu _{m} \ : \ m \ge \ell \} \end{aligned}$$
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 equationsIt 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).
$$\begin{aligned} \left\{ \begin{array}{ll} Y^{q+1}=X^q+X, &{} \\ Z^{q^2-q+1}=Y^{q^2}-Y. &{} \end{array} \right. \end{aligned}$$
(2)
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 isEach 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
$$\begin{aligned} \begin{aligned} G(P)&= \{ iq^3 + kq + m(q^2 + 1) + \sum ^{q-2}_{s=1} ( n_s(s + 1)q^2) \\&\quad +\, j + 1 \,\mid \, i, j, k,m,\ldots , n_{q-2} \ge 0, \\ j&\le q-1,\text { and }i + j + k + mq + \sum ^{q-2}_{s=1} (n_s((s + 1)q - s)) \le q^2 - 2\}. \end{aligned} \end{aligned}$$
(3)
-
\(G_1\) 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 isWe use an analogous notation for the elements of \(Ap_{2,1}\), \(Ap_{2,2}\), \(Ap_3\) and \(Ap_4\).\(\square\)
$$\begin{aligned} x=a(q^3-1)+i(q^3-q)+j(q^4-q^3-q^2). \end{aligned}$$
-
\(Ap_1\cap Ap_{2,1}\) 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 \(x_{a,i,j}=x_{\bar{i},\bar{j}}\) thenReducing 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$$\begin{aligned}&a(q^3-1)+i(q^3-q)+j(q^4-q^3-q^2)\nonumber \\&\quad = (q^3-1)+\bar{i}(q^3-q)+\bar{j}(q^4-q^2-1). \end{aligned}$$(4)whence \(j=\bar{j}\), a contradiction since \(j\le a-2=\bar{j}-1\).$$\begin{aligned} j(q^2-q-1)=\bar{j}(q^2-q-1), \end{aligned}$$
-
Let \(x_{a,i,j} \in Ap_1\) and \(x_{\bar{i},\bar{j}} \in Ap_{3}\). If \(x_{a,i,j}=x_{\bar{i},\bar{j}}\) thenthat modulo q yields \(a=1\), a contradiction with \(a\ge 2\).$$\begin{aligned}&a(q^3-1)+i(q^3-q)+j(q^4-q^3-q^2)\\&\quad = 2q^3-1+\bar{i}(q^3-q)+\bar{j}(q^4-q^3-q^2), \end{aligned}$$
-
Let \(x_{a,i,j} \in Ap_1\) and \(x_{\bar{a},\bar{i},\bar{j}} \in Ap_{4}\). If \(x_{a,i,j}=x_{\bar{a},\bar{i},\bar{j}}\) thenthat modulo q yields \(a=\bar{a}\). Therefore$$\begin{aligned}&a(q^3-1)+i(q^3-q)+j(q^4-q^3-q^2)\\&\quad = q^3+\bar{a}(q^3-1)+\bar{i}(q^3-q)+\bar{j}(q^4-q^3-q^2), \end{aligned}$$whence \(i=\bar{i}\) follows. Thus$$\begin{aligned} i(q^3-q)+j(q^4-q^3-q^2)= q^3+\bar{i}(q^3-q)+\bar{j}(q^4-q^3-q^2), \end{aligned}$$whence \(j\ge \bar{j}\), a contradiction with \(j\le a-2=\bar{a}-2\le \bar{j}-2\).$$\begin{aligned} j(q^4-q^3-q^2)=q^3+\bar{j}(q^4-q^3-q^2), \end{aligned}$$
-
\(Ap_{2,2}\cap Ap_{3}\) 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 \(x_{i,j}=\bar{x}_{\bar{a},\bar{i},\bar{j}}\) thenwhence reducing modulo q yields \(j=\bar{a}-1\). Now Equation (6) reads$$\begin{aligned}&(q^3-1)+i(q^3-q)+j(q^4-q^2-1)\nonumber \\&\quad = q^3+\bar{a}(q^3-1)+\bar{i}(q^3-q)+\bar{j}(q^4-q^3-q^2), \end{aligned}$$(5)and hence \(i=\bar{i}\). Therefore$$\begin{aligned} i(q^2-1)+j(q^3-q^2-q)-q^2= \bar{i}(q^2-1)+\bar{j}(q^3-q^2-q), \end{aligned}$$(6)and \(j\ge \bar{j}\), a contradiction with \(j=\bar{a}-1\le \bar{j}-1\).$$\begin{aligned} j(q^3-q^2-q)=q^2+\bar{j}(q^3-q^2-q) \end{aligned}$$
-
\(Ap_3\cap Ap_{4}\) is empty since for every element x of \(Ap_3\), \(x+1\) is divisible by q, but this fails for any element of \(Ap_4\).
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 \(x,y \in Ap_1\), soandwith \(a,\bar{a}\in \{2,\dots ,q-1\}\), \(i,\bar{i}\in \{0,\dots , q-1\}\), 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 \(a=\bar{a}\). ThereforewhenceBy applying the same argument as above, we obtain \(i=\bar{i}\). Finally, this implies \(j=\bar{j}\), and so the claim follows. \(\square\)
$$\begin{aligned} x=a(q^3-1)+i(q^3-q)+j(q^4-q^3-q^2) \end{aligned}$$
$$\begin{aligned} y=\bar{a}(q^3-1)+\bar{i}(q^3-q)+\bar{j}(q^4-q^3-q^2), \end{aligned}$$
$$\begin{aligned} i(q^3-q)+j(q^4-q^3-q^2)=\bar{i}(q^3-q)+\bar{j}(q^4-q^3-q^2), \end{aligned}$$
$$\begin{aligned} i(q^2-1)+j(q^3-q^2-q)=\bar{i}(q^2-1)+\bar{j}(q^3-q^2-q). \end{aligned}$$
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, \(\square\)
(a)
if \(x \in Ap_1\) then where \(a\in \{2,\dots ,q-1\}\), \(i\in \{0,\dots , q-1\}\) and \(j\in \{0,\dots , a-2\}\). Therefore Therefore \(x-q^3\not \in H(P)\) by (3).
$$\begin{aligned}&x-q^3=a(q^3-1)+i(q^3-q)+j(q^4-q^3-q^2)-q^3\\&\quad =(a+i+jq-j-2)q^3+(q-j-1)q^2+(q-i-1)q+q-a-1 +1, \end{aligned}$$
$$\begin{aligned} {\left\{ \begin{array}{ll} a+i+jq-j-2\ge 0\\ q-j-1\ge 0 \\ q-i-1\ge 0 \\ 0\le q-a-1\le q-1\\ (a+i+jq-j-2)+q(q-j-1)-(q-j-2)+(q-i-1)+\\ +(q-a-1)=q^2-2.\\ \end{array}\right. } \end{aligned}$$
(b)
if \(x \in Ap_{2,1}{\setminus } \{q^3\}\) then 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 Therefore \(x-q^3\not \in H(P)\) by (3).
$$\begin{aligned} x-q^3= & {} i(q^3-q)+j(q^4-q^3-q^2)\nonumber \\= & {} (i+jq-j-1)q^3+(q-j-1)(q^2+1)+(q-i-1)q+j+1; \end{aligned}$$
(7)
$$\begin{aligned} {\left\{ \begin{array}{ll} i+jq-j-1\ge 0\\ q-j-1\ge 0 \\ q-i-1\ge 0\\ 0\le j\le q-1\\ i+jq-j-1+q(q-j-1)+(q-i-1)+j=q^2-2.\\ \end{array}\right. } \end{aligned}$$
(c)
if \(x \in Ap_{2,2}\) then 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 Therefore \(x-q^3\not \in H(P)\) by (3).
$$\begin{aligned} x-q^3= & {} i(q^3-q)+j(q^4-q^2-1)-1\nonumber \\= & {} (i+jq-1)q^3+(q-j-2)(q^2+1)+(2q-i-1)q+1; \end{aligned}$$
(8)
$$\begin{aligned} {\left\{ \begin{array}{ll} i+jq-1\ge 0 \\ q-j-2\ge 0\\ 2q-i-1\ge 0\\ i+jq-1+q(q-j-2)+(2q-i-1)=q^2-2.\\ \end{array}\right. } \end{aligned}$$
(d)
if \(x \in Ap_3\) then where \(i\in \{0,\dots , q-1\}\) and \(j\in \{1,\dots , q-1\}\). Therefore Therefore \(x-q^3\not \in H(P)\) by (3).
$$\begin{aligned}&x-q^3= q^3-1+i(q^3-q)+j(q^4-q^3-q^2)\\&\quad =(i+jq-j)q^3+(q-j-1)(q^2+1)+(q-i-1)q+ j-1+1; \end{aligned}$$
$$\begin{aligned} {\left\{ \begin{array}{ll} i+jq-j\ge 0\\ q-j-1\ge 0\\ q-i-1\ge 0\\ 0\le j-1 \le q-1\\ i+jq-j+q(q-j-1)+(q-i-1)+j-1=q^2-2.\\ \end{array}\right. } \end{aligned}$$
(e)
if \(x \in Ap_4\) then where \(i\in \{0,\dots , q-1\}\), \(j\in \{2,\dots , q-1\}\) and \(a\in \{2,\dots ,j\}\). Therefore Therefore \(x-q^3\not \in H(P)\) by (3).
$$\begin{aligned}&x-q^3= a(q^3-1)+i(q^3-q)+j(q^4-q^3-q^2)\\&\quad =(i+jq-j+a-1)q^3\\&\quad \quad +(q-j-1)(q^2+1)+(q-i-1)q+ j-a+1; \end{aligned}$$
$$\begin{aligned} {\left\{ \begin{array}{ll} i+jq-j+a-1\ge 0\\ q-j-1\ge 0\\ q-i-1\ge 0\\ 0\le j-a\le q-1\\ i+jq-j+a-1+q(q-j-1)+(q-i-1)+j-a=q^2-2.\\ \end{array}\right. } \end{aligned}$$
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 \(x\in S_1\). Thenfor some \(0\le i\le q-1\) and \(0\le j \le q-1\). It may be observed thatWe 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 \(x \in G_3\).\(\square\)
$$\begin{aligned} x=q^3+i(q^3-q)+j(q^4-q^3-q^2), \end{aligned}$$
$$\begin{aligned} x=(i+jq-j)q^3+(q-j-1)(q^2+1)+(q-i-1)q+j+1. \end{aligned}$$
-
Case \(x\in G_1\). There exist non-negative integers \(\bar{m},\bar{i},\bar{k}\) such that \(\bar{i}+\bar{k}+\bar{m}q\le q^2-2\) andEquation (9) modulo q yields$$\begin{aligned}&(i+jq-j)q^3+(q-j-1)(q^2+1)+(q-i-1)q+j+1\nonumber \\&\quad =\bar{i}q^3+\bar{m}(q^2+1)+\bar{k}q+1. \end{aligned}$$(9)whence \(\bar{m}=q-1\). Hence$$\begin{aligned} \bar{m}\equiv -1 \pmod q, \end{aligned}$$and, dividing by q,$$\begin{aligned} (i+jq-j)q^3+(q-j-1)q^2+(q-i-1)q=\bar{i}q^3+\bar{m}q^2+\bar{k}q, \end{aligned}$$that is$$\begin{aligned} (i+jq-j)q^2+(q-j-1)q+q-i-1=\bar{i}q^2+(q-1)q+\bar{k}, \end{aligned}$$Equation (10) modulo q now yields$$\begin{aligned} (i+jq-j)q^2-jq+q-i-1=\bar{i}q^2+\bar{k}. \end{aligned}$$(10)Moreover \(\bar{i}+\bar{k}+\bar{m}q\le q^2-2\), gives \(\bar{k}+\bar{i}\le q-2\) and hence \(\bar{k}=q-i-1\).$$\begin{aligned} \bar{k} \equiv -i-1 \pmod q. \end{aligned}$$Substituting in Eq. (10) we obtainAgain dividing by q and reducing shows \(j\equiv 0 \pmod q\), whence \(j=0\). Therefore \(\bar{i}=i\), and a contradiction arises from \(\bar{k}+\bar{i}\le q-2\).$$\begin{aligned} (i+jq-j)q^2-jq=\bar{i}q^2. \end{aligned}$$
-
Case \(x\in G_2\). There exist non-negative integers \(\bar{m},\bar{i},\bar{k}\) and \(\bar{j}\) such thatand$$\begin{aligned} {\left\{ \begin{array}{ll} 1\le \bar{j}\le q-1,\\ \bar{k}\le q-1,\\ \bar{j}+\bar{m}\le q-1,\\ \bar{i}+\bar{k}+\bar{j}+\bar{m}q\le q^2-2\\ \end{array}\right. } \end{aligned}$$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$$\begin{aligned}&(i+jq-j)q^3+(q-j-1)(q^2+1)+(q-i-1)q+j+1\nonumber \\&\quad =\bar{i}q^3+\bar{m}(q^2+1)+\bar{k}q+\bar{j}+1. \end{aligned}$$(11)that is$$\begin{aligned}&(i+jq-j)q^3+(q-j-1)(q^2+1)+(q-i-1)q+j+1\\&\quad =\bar{i}q^3+\bar{m}q^2+\bar{k}q+\bar{m}+\bar{j}+1, \end{aligned}$$Again, \(\bar{k}\le q-1\) and Eq. (12) modulo q imply \(\bar{k}=q-i-1\). Thus$$\begin{aligned} (i+jq-j)q^2+(q-j-1)q+q-i-1=\bar{i}q^2+\bar{m}q+\bar{k}. \end{aligned}$$(12)whence \(\bar{m}=q-j-1\) and \(\bar{j}=j\). Finally, \(\bar{i}=i+jq-j\) and$$\begin{aligned} (i+jq-j)q+(q-j-1)=\bar{i}q+\bar{m}, \end{aligned}$$(13)a contradiction.$$\begin{aligned}&\bar{i}+\bar{k}+\bar{j}+\bar{m}q=i+jq-j+q-i-1\\&\quad + j+(q-j-1)q = q^2-1>q^2-2, \end{aligned}$$
-
Case \(x\in G_3\). There exist non-negative integers \(s,\bar{i},\bar{k}\) and \(\bar{j}\) such thatand$$\begin{aligned} {\left\{ \begin{array}{ll} 1\le s\le q-2,\\ \bar{j},\bar{k}\le q-1,\\ \bar{i}+\bar{k}+(s+1)q\ge q^2-1,\\ \bar{i}+\bar{j}+\bar{k}+(s+1)q-s\le q^2-2\\ \end{array}\right. } \end{aligned}$$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.$$\begin{aligned}&(i+jq-j)q^3+(q-j-1)(q^2+1)+(q-i-1)q+j+1\nonumber \\&\quad =\bar{i}q^3+(s+1)q^2+\bar{k}q+\bar{j}+1. \end{aligned}$$(14)
Proposition 5
\(A=Ap(H(P),q^3)=Ap(T,q^3)\).
Proof
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 \(C_{\ell }(P) =C^{\perp }(D,\rho _{\ell }P)\), whereis 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 \(n=q^7\), \(k=n-\ell\) andwhere \(d^*=\deg (G)-2\mathfrak {g}+2\) 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.
$$\begin{aligned} D=\sum _{Q\in {\mathcal {X}}(\mathbb {F}_{q^7}) {\setminus }{\{P\}} } Q \end{aligned}$$
$$\begin{aligned} d\ge \max \{d_{ORD}(C_{\ell }(P)),d^*\}, \end{aligned}$$
(15)
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 haveHence Proposition 6 yieldsIn 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 \(d^*\).
$$\begin{aligned} \rho _{\ell }=2\mathfrak {g}+(2\mathfrak {g}-2q^2+r+2)=4\mathfrak {g}-2q^2+r+2. \end{aligned}$$
$$\begin{aligned} d_{ORD}(C_{\ell }(P))=\ell +1-\mathfrak {g}=2\mathfrak {g}-2q^2+4+r=\rho _\ell -2\mathfrak {g}+2=d^*. \end{aligned}$$
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 \(C_i\), thenwhere \(h_i\) 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 \(n=q^7\) andIt may be noted thatwhere \(d_1\) is the minimum distance of \(C(D,\rho _{\ell +s}P)\).
$$\begin{aligned} k_2=q^7-h_{\ell } \quad \text { and } \quad k_1=q^7 -h_{\ell +s}=q^7-h_\ell -s, \end{aligned}$$
$$\begin{aligned} d =\min \lbrace w(c)\, \vert \, c \in (C_\ell {\setminus } C_{\ell +s})\cup (C(D,\rho _{\ell +s}P){\setminus } C(D,\rho _\ell P)) \rbrace . \end{aligned}$$
$$\begin{aligned} d\ge \min \lbrace d_{ORD}(C_\ell ),d_1 \rbrace , \end{aligned}$$
(16)
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,and therefore it goes to 0 as q goes to infinity.
$$\begin{aligned} \varDelta _Q=\frac{q^7-s-2d+2}{q^7}=\frac{2\ell -2d+2}{q^7}\le \frac{2\mathfrak {g}}{q^7}=\frac{q^5-2q^3+q^2}{q^7}, \end{aligned}$$
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 thatThen there exists a \([[n,k,d]]_q\)-quantum code with \(k=\sum _{i=1}^t b_i-\sum _{i=1}^t a_i\) and \(d\ge \min \lbrace n-\sum _{i=1}^t b_i, \sum _{i=1}^t a_i-(2\mathfrak {g}-2)\rbrace\).
$$\begin{aligned} 2\mathfrak {g}-2<\sum _{i=1}^t a_i<\sum _{i=1}^t b_i < n. \end{aligned}$$
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 thatThen there exists a quantum code with parameters \([[q^7,b-a,d]]_{q^7}\), where
$$\begin{aligned} q^5-2q^3+q^2-2<a<b<q^7. \end{aligned}$$
$$\begin{aligned} d\ge \min \lbrace q^7-b,\; a-(q^5-2q^3+q^2-2) \rbrace . \end{aligned}$$
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”.
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