Weierstrass Semigroup and Pure Gaps at several points on the $GK$ curve

TL;DR

This paper characterizes the Weierstrass semigroup at multiple points on the GK curve, identifies pure gaps, and applies these findings to construct algebraic geometry codes with improved parameters.

Contribution

It provides explicit descriptions of the Weierstrass semigroup and pure gaps at multiple points on the GK curve, enabling better code construction.

Findings

Explicit description of Weierstrass semigroup at multiple points

Conditions for identifying pure gaps on the GK curve

Construction of algebraic geometry codes with improved parameters

Abstract

We determine the Weierstrass semigroup $H(P_{\infty}, P_{1}, \ldots , P_{m})$ at several points on the $GK$ curve. In addition, we present conditions to find pure gaps on the set of gaps $G(P_{\infty}, P_{1}, \ldots , P_{m})$. Finally, we apply the results to obtain AG codes with good relative parameters.

Equations

$$H(P_1,\dots ,P_m)=\{(a_1,\dots ,a_m)\in {\mathbb{N}}_0^m\text{mbox};\exists f\in {\mathbb{F}}_q(\mathcal{X})\text{mbox}with(f{)}_{\infty }=\sum _{i=1}^m{a}_i{P}_i\}$$

$$lub\{{\mathbf{u}}_1,\dots ,{\mathbf{u}}_t\}=(\max \{u_{1_1},\dots ,u_{t_1}\},\dots ,\max \{u_{1_m},\dots ,u_{t_m}\})\in {\mathbb{N}}_0^m.$$

$${\nabla }_i(\mathbf{n}):=\{(p_1,\dots ,p_m)\in H(P_1,\dots ,P_m)\text{mbox};p_i=n_i\}.$$

$$\Gamma (P_1,\dots ,P_m):=\{\mathbf{n}\in {\mathbb{N}}^m:\text{mbox}forsomei,1\le i\le m,\mathbf{n}\text{mbox}isminimalin{\nabla }_i(\mathbf{n})\}.$$

$$\left\{\begin{array}[]{cl}lub\{{\bf u}_{1},\ldots,{\bf u}_{m}\}\in\mathbb{N}_{0}^{m}:&{\bf u}_{i}\in\Gamma(P_{1},\ldots,P_{m})\\ &\mbox{ or }(u_{i_{1}},\ldots,u_{i_{k}})\in\Gamma(P_{i_{1}},\ldots,P_{i_{k}})\\ &\mbox{ for some }\{i_{1},\ldots,i_{k}\}\subset\{1,\ldots,m\}\mbox{ such that }\\ &i_{1}<\cdots<i_{k}\mbox{ and }u_{i_{k+1}}=\cdots=u_{i_{\ell}}=0,\\ &\mbox{ where }\{i_{k+1},\ldots,i_{m}\}\subset\{1,\ldots,\ell\}\setminus\{i_{1},\ldots,i_{k}\}\end{array}\right\}\;.$$

$$\begin{array}[]{ccl}C_{\mathcal{L}}(D,G)&:=&\{(f(P_{1}),\ldots,f(P_{n}))\mbox{ ; }f\in L(G)\};\\ C_{\Omega}(D,G)&:=&\{(res_{P_{1}}(\eta),\ldots,res_{P_{n}}(\eta))\mbox{ ; }\eta\in\Omega(G-D)\}.\end{array}$$

$$\left\{\begin{array}[]{c}Z^{n^{2}-n+1}=Yh(X)\\ X^{n}+X=Y^{n+1}\;\;\;,\end{array}\right.$$

$$\begin{array}[]{rcl}(x-a_{j})&=&(n^{3}+1)P_{j}-(n^{3}+1)P_{\infty},\mbox{ for }j=1,\ldots,n;\\ (y)&=&\sum_{j=1}^{n}(n^{2}-n+1)P_{j}-n(n^{2}-n+1)P_{\infty};\\ (z)&=&\sum_{j=1}^{n}P_{j}+\sum_{\ell=1}^{n^{3}-n}Q_{\ell}-n^{3}P_{\infty}.\end{array}$$

$$c{P}_j\sim c{P}_{\infty },$$

$$a{P}_1+\cdots +a{P}_n\sim na{P}_{\infty },$$

$$P_1+\cdots +P_n+Q_1+\cdots +Q_{n^3-n}\sim b{P}_{\infty }.$$

$$\left(n^2-m-\sum _{s=1}^m{j}_s\right)c-ina-kb>0.$$

$$A^{\prime }=((n^2-m)c-ina-kb)P_{\infty }+\sum _{s=1}^m(ia+k)P_s$$

$$((n^2-m-\sum _{s=1}^m{j}_s)c-ina-kb)P_{\infty }+\sum _{s=1}^m(j_{s}c+ia+k)P_s\sim A^{\prime }.$$

$$\begin{array}[]{ll}K+P+Q-A^{\prime}&\displaystyle=(n^{2}-2)cP_{\infty}+P+Q-((n^{2}-m)c-ina-kb)P_{\infty}-\sum_{s=1}^{m}(ia+k)P_{s}\\ &\displaystyle=(c(m-2)+ina+kb)P_{\infty}+P+Q-\sum_{s=1}^{m}(ia+k)P_{s}\end{array}$$

$$K+P+Q-A^{\prime }=(c(m-1)+ina+(k-1)b)P_{\infty }-(ia+k-1)P_1-\sum _{s=2}^m(ia+k)P_s$$

$$z^{k-1}{y}^i(x-a_2)\cdots (x-a_m)\in \mathcal{L}(K+P+Q-A^{\prime })\setminus \mathcal{L}(K+Q-A^{\prime }).$$

$$z^{k-1}{y}^i(x-a_3)\cdots (x-a_m)\in \mathcal{L}(K+P+Q-A^{\prime })\setminus \mathcal{L}(K+Q-A^{\prime }).$$

$$\begin{array}[]{ll}\Gamma_{m+1}&=\left\{\left((n^{2}-m-\sum_{s=1}^{m}j_{s})c-ina-kb,j_{1}c+ia+k,\ldots,j_{m}c+ia+k\right)\right.;\\ &\left.1\leq k\leq a\mbox{, }0\leq i\leq n\mbox{, }j_{s}\geq 0\mbox{ and }\left(n^{2}-m-\sum_{s=1}^{m}j_{s}\right)c-ina-kb>0\right\}.\end{array}$$

$$f=\frac{y^{n-i}{z}^{a-k}}{(x-a_1{)}^{j_1+1}\dots (x-a_m{)}^{j_m+1}}.$$

$$\left\{\begin{array}[]{c}Z^{3}=Y(1+X+X^{2})\\ X^{2}+X=Y^{3}\end{array}\right.$$

$$\begin{array}[]{ccl}(x-a_{\ell})&=&9P_{\ell}-9P_{\infty}\,;\\ (y)&=&3P_{1}+3P_{2}-6P_{\infty}\,;\\ (z)&=&P_{1}+P_{2}+Q_{1}+\cdots+Q_{6}-8P_{\infty}\,,\end{array}$$

$$\Gamma (P_{\infty },P_1)=\{(1,19),(2,11),(3,3),(4,13),(5,5),(7,7),(10,10),(11,2),(13,4),(19,1)\}.$$

$$\Gamma (P_{\infty },P_1,P_2)=\{(9(2-j_1-j_2)-6i-8k,9j_1+3i+k,9j_2+3i+k):$$

$$0\le i\le 2,1\le k\le 3,j_1,j_2\ge 0\text{mbox}and9(2-j_1-j_2)-6i-8k>0\}$$

$$\Gamma (P_{\infty },P_1,P_2)=\{(10,1,1),(1,1,10),(1,10,1),(2,2,2),(4,4,4)\}.$$

$$Z^7=Y(2+X^2+2X^4+X^6),X^3+X=Y^4.$$

Full text

1. Introduction

Curves with many rational points and a large automorphism group have been investigated for their applications in coding theory. In [9], Giulietti and Korchmáros introduced a maximal curve, so called $GK$ curve, which is not a subcover of the corresponding Hermitian curve. The $GK$ curve is one of the rare examples of curves over a finite field where the automorphism group $Aut(GK)$ is rather large with respect to the genus. Another interesting fact about this curve is that the set of rational points splits into two non-trivial orbits, $\mathcal{O}_{1}$ and $\mathcal{O}_{2}$, and $Aut(GK)$ acts on $\mathcal{O}_{1}$ as $PGU(3,n)$ in its doubly transitive permutation representation, see [6, Theorem 3.4]. More recently, we can find applications of the $GK$ curve in coding theory, see [1], [3] and [6].

As is known, Weierstrass semigroup is also an important tool in coding theory, see e.g. [2], [8] and [13]. In this work, we determine the Weierstrass semigroup $H(P_{\infty},P_{1},\ldots,P_{m})$ at $m+1$ points on $\mathcal{O}_{1}$, with $1\leq m\leq|\mathcal{O}_{1}|$, where $P_{\infty}$ is the single point at infinity on $GK$. Our results were obtained using the concept of discrepancy, for given rational points $P$ and $Q$ on a curve $\mathcal{X}$, see Definition 2.5. This concept was introduced by Duursma and Park in [5], and it was our main tool for obtain the set $\Gamma(P_{\infty},P_{1},\ldots,P_{m})$, called minimal generating set of $H(P_{\infty},P_{1},\ldots,P_{m})$, see Theorem 3.4. In addition, we present conditions to find pure gaps on the set of gaps $G(P_{\infty},P_{1},\ldots,P_{m})$.

This paper is organized as follows. Section 2 contains general results about Weierstrass semigroup and discrepancy, in addition to basic facts about AG codes and the $GK$ curve. In Section 3, we determine the minimal generating set for the Weierstrass semigroup $H(P_{\infty},P_{1},\ldots,P_{m})$ at points on the orbit $\mathcal{O}_{1}$ cited above. Finally, in Section 4 we present some results about pure gaps and AG codes over the $GK$ curve.

2. Preliminaries

We begin this section by introducing some notations that will be used in this work. Let $\mathcal{X}$ be a nonsingular, projective, geometrically irreducible curve of genus $g\geq 1$ defined over a finite field $\mathbb{F}_{q}$, let $\mathbb{F}_{q}(\mathcal{X})$ be the field of rational functions and $Div(\mathcal{X})$ be the set of divisors on $\mathcal{X}$. For $f\in\mathbb{F}_{q}(\mathcal{X})$, the divisor of $f$ will be denoted by $(f)$ and the divisor of poles of $f$ by $(f)_{\infty}$. For a divisor $G$ on $\mathcal{X}$, let $\mathcal{L}(G):=\{f\in\mathbb{F}_{q}(\mathcal{X})\mbox{ ; }(f)+G\geq 0\}\cup\{0\}$ be the Riemann-Roch space of $G$ and let $\dim(\mathcal{L}(G))$ be the dimension of $\mathcal{L}(G)$ as an $\mathbb{F}_{q}$-vector space. Let $\Omega(G)$ be the space of differentials $\eta$ on $\mathcal{X}$ such that $\eta=0$ or $div(\eta)\geq G$, where $div(\eta)=\sum_{P\in\mathcal{X}}ord_{P}(\eta)P$ and $ord_{P}(\eta)$ is the order of $\eta$ at $P$. As follows, we denote $\mathbb{N}_{0}=\mathbb{N}\cup\{0\}$, where $\mathbb{N}$ is the set of positive integers.

2.1. Weierstrass semigroup and Discrepancy

Let $P_{1},\ldots,P_{m}$ be distinct rational points on $\mathcal{X}$. The set

[TABLE]

is called the Weierstrass semigroup at the points $P_{1},\ldots,P_{m}$. It is not difficult to see that the set $H(P_{1},\ldots,P_{m})$ is a semigroup. An element in $\mathbb{N}_{0}^{m}\setminus H(P_{1},\ldots,P_{m})$ is called gap and the set $G(P_{1},\ldots,P_{m})=\mathbb{N}_{0}^{m}\setminus H(P_{1},\ldots,P_{m})$ is called gap set of $P_{1},\ldots,P_{m}$.

Define a partial order $\preceq$ on $\mathbb{N}_{0}^{m}$ by $(n_{1},\ldots,n_{m})\preceq(p_{1},\ldots,p_{m})$ if and only if $n_{i}\leq p_{i}$ for all $i$, $1\leq i\leq m$.

For ${\bf u}_{1},\ldots,{\bf u}_{t}\in\mathbb{N}_{0}^{m}$, where, for all $k$, ${\bf u}_{k}=(u_{k_{1}},\ldots,u_{k_{m}})$, we define the least upper bound ($lub$) of the vectors ${\bf u}_{1},\ldots,{\bf u}_{t}$ in the following way:

[TABLE]

For $\mathbf{n}=(n_{1},\ldots,n_{m})\in\mathbb{N}_{0}^{m}$ and $i\in\{1,\ldots,m\}$, we set

[TABLE]

Proposition 2.1**.**

[12, Proposition 3]** Let $\mathbf{n}=(n_{1},\ldots,n_{m})\in\mathbb{N}_{0}^{m}$. Then $\mathbf{n}$ is minimal, with respect to $\preceq$, in $\nabla_{i}(\mathbf{n})$ for some $i$, $1\leq i\leq m$, if and only if $\mathbf{n}$ is minimal in $\nabla_{i}(\mathbf{n})$ for all $i$, $1\leq i\leq m$.

Proposition 2.2**.**

[12, Proposition 6]** Suppose that $1\leq t\leq m\leq q$ and ${\bf u}_{1},\ldots,{\bf u}_{t}\in H(P_{1},\ldots,P_{m})$. Then $lub\{{\bf u}_{1},\ldots,{\bf u}_{t}\}\in H(P_{1},\ldots,P_{m})$.

Definition 2.3**.**

Let $\Gamma(P_{1})=H(P_{1})$ and, for $m\geq 2$, define

[TABLE]

Lemma 2.4**.**

[12, Lemma 4]** For $m\geq 2$, $\Gamma(P_{1},\ldots,P_{m})\subseteq G(P_{1})\times\cdots\times G(P_{m})\;.$

In [12], Theorem 7, it is shown that, if $2\leq m\leq q$, then $H(P_{1},\ldots,P_{m})=$

[TABLE]

Therefore, the Weierstrass semigroup $H(P_{1},\ldots,P_{m})$ is completely determined by $\Gamma(P_{1},\ldots,P_{m})$. In [12], Matthews called the set $\Gamma(P_{1},\ldots,P_{m})$ of minimal generating set of $H(P_{1},\ldots,P_{m})$.

In [5, Section 5], Duursma and Park introduced the concept of discrepancy as follows.

Definition 2.5**.**

A divisor $A\in Div(\mathcal{X})$ is called a discrepancy for two rational points $P$ and $Q$ on $\mathcal{X}$ if $\mathcal{L}(A)\neq\mathcal{L}(A-P)=\mathcal{L}(A-P-Q)$ and $\mathcal{L}(A)\neq\mathcal{L}(A-Q)=\mathcal{L}(A-P-Q)$.

The next result relates the concept of discrepancy with the set $\Gamma(P_{1},\ldots,P_{m})$.

Lemma 2.6**.**

[4, Lemma 2.6]** Let ${\bf n}=(n_{1},\ldots,n_{m})\in H(P_{1},\ldots,P_{m})$. Then ${\bf n}\in\Gamma(P_{1},\ldots,P_{m})$ if and only if the divisor $A=n_{1}P_{1}+\cdots+n_{m}P_{m}$ is a discrepancy with respect to $P$ and $Q$ for any two rational points $P,Q\in\{P_{1},\ldots,P_{m}\}$.

2.2. AG codes

Let $D=P_{1}+\ldots+P_{n}$ be a divisor on $\mathcal{X}$ such that $P_{i}\neq P_{j}$ for $i\neq j$. Let $G$ be another divisor on $\mathcal{X}$ such that $supp(D)\cap supp(G)=\emptyset$. Consider the maps $e_{v}:\mathcal{L}(G)\rightarrow\mathbb{F}_{q}^{n}$ and $\varphi:\Omega(G-D)\rightarrow\mathbb{F}_{q}^{n}$ defined, respectively, by $e_{v}(f):=(f(P_{1}),\ldots,f(P_{n}))$ and $\varphi(\eta):=(res_{P_{1}}(\eta),\ldots,res_{P_{n}}(\eta))$, where $res_{P_{i}}(\eta)$ is the residue of $\eta$ at $P_{i}$, $i=1,\ldots,n$. We define the AG codes $C_{\mathcal{L}}(D,G)$ and $C_{\Omega}(D,G)$ as the images of the maps $e_{v}$ and $\varphi$, respectively. That is,

[TABLE]

The AG codes $C_{\mathcal{L}}(D,G)$ and $C_{\Omega}(D,G)$ are dual to each other. Let $[n,k,d]$ and $[n,k_{\Omega},d_{\Omega}]$ be the length, dimension and minimum distance of $C_{\mathcal{L}}(D,G)$ and $C_{\Omega}(D,G)$, respectively. By Riemann-Roch Theorem we can estimate the parameters $[n,k,d]$ and $[n,k_{\Omega},d_{\Omega}]$. In particular, if $2g-2<deg(G)<n$, we have that $k=deg(G)-g+1$, $d\geq n-deg(G)$, $k_{\Omega}=n-deg(G)+g-1$ and $d_{\Omega}\geq deg(G)-2g+2$, see e.g. [15]. The right-hand side of the inequalities involving the minimum distance is known as the Goppa bound. One of the ways to obtain codes with good parameters is to find codes whose minimum distance have bounds better than the Goppa bound. In addition, another way is study codes over curves with many rational points, more specifically, codes arising from maximal curves. We remember that a curve $\mathcal{X}$ of genus $g$ over $\mathbb{F}_{q}$ is a maximal curve if its number of rational points attains the Hasse-Weil upper bound, namely equals $2g\sqrt{q}+q+1$.

If $G=aQ$ for some rational point $Q$ on $\mathcal{X}$ and $D$ is the sum of all the other rational points on $\mathcal{X}$, then the AG codes $C_{\mathcal{L}}(D,G)$ and $C_{\Omega}(D,G)$ are called one-point AG codes. Analogously, if $G=a_{1}Q_{1}+\cdots+a_{m}Q_{m}$, for $m$ distinct rational points on $\mathcal{X}$ and $D$ is the sum of all the other rational points on $\mathcal{X}$, then $C_{\mathcal{L}}(D,G)$ and $C_{\Omega}(D,G)$ are called $m$-point AG codes. For more details about coding theory, see [10], [14] and [15].

2.3. The $GK$ curve

Let $q=n^{3}$, where $n\geq 2$ is a prime power. The $GK$ curve over $\mathbb{F}_{q^{2}}$ is the curve of $\mathbb{P}^{3}(\overline{\mathbb{F}}_{q^{2}})$ with affine equations

[TABLE]

where $\displaystyle h(X)=\sum_{i=0}^{n}(-1)^{i+1}X^{i(n-1)}$. We will denote this curve simply by $GK$. The curve $GK$ is absolutely irreducible, nonsingular, has $n^{8}-n^{6}+n^{5}+1$ $\mathbb{F}_{q^{2}}$-rational points, a single point at infinity $P_{\infty}=(1:0:0:0)$ and its genus is $g=\dfrac{1}{2}(n^{3}+1)(n^{2}-2)+1$. The $GK$ curve has an important properties as it lies on the Hermitian surface $\mathcal{H}_{3}$ with affine equation $X^{n^{3}}+X=Y^{n^{3}+1}+Z^{n^{3}+1}$; it is a maximal curve and, for $q>8$, $GK$ is the only know curve that is maximal but not $\mathbb{F}_{q^{2}}$-covered by the Hermitian curve $\mathcal{H}_{2}$ defined over $\mathbb{F}_{q^{2}}$ and its automorphism group $Aut(GK)$ has size $n^{3}(n^{3}+1)(n^{2}-1)(n^{2}-n+1)$ which turns out to be very large compared to the genus $g$.

Let $GK(\mathbb{F}_{q^{2}})$ be the set of $\mathbb{F}_{q^{2}}$-rational points of $GK$. For $j=1,\ldots,n$, let $P_{j}=(a_{j},0,0)\in GK(\mathbb{F}_{q^{2}})$ such that $a_{j}^{n}+a_{j}=0$, and, for $\ell=1,\ldots,n^{3}-n$, let $Q_{\ell}=(a_{\ell},b_{\ell},0)\in GK(\mathbb{F}_{q^{2}})$ such that $b_{\ell}\neq 0$ and $a_{\ell}^{n}+a_{\ell}=b_{\ell}^{n+1}$. In the following, $P_{\infty}$, $P_{j}$, for $j=1,\ldots,n$, and $Q_{\ell}$, for $\ell=1,\ldots,n^{3}-n$, will be the points given above.

Since $P_{\infty}$ is a single point at infinity of $GK$ and the function field $\mathbb{F}_{q^{2}}(GK)$ is $\mathbb{F}_{q^{2}}(x,y,z)$ with $z^{n^{2}-n+1}=yh(x)$ and $x^{n}+x=y^{n+1}$ we have that

[TABLE]

Theorem 2.7**.**

[6, Theorem 3.4]** The set of $\mathbb{F}_{q^{2}}$-rational points of $GK$ splits into two orbits under the action of $Aut(GK)$. One orbit, say $\mathcal{O}_{1}$, has size $n^{3}+1$ and consists of the points $P_{j}$ and $Q_{\ell}$ as above together with the infinite point $P_{\infty}$. The other orbit has size $n^{3}(n^{3}+1)(n^{2}-1)$ and consists of the points $P=(a,b,c)\in GK(\mathbb{F}_{q^{2}})$ with $c\neq 0$. Furthermore, $Aut(GK)$ acts on $\mathcal{O}_{1}$ as $PGU(3,n)$ in its doubly transitive permutation representation.

Proposition 2.8**.**

[6, Proposition 3.1]** Let $P_{j}$ and $Q_{\ell}$ be as above. Then, $H(P_{\infty})=H(P_{j})=H(Q_{\ell})=\langle n^{3}-n^{2}+n,n^{3},n^{3}+1\rangle$, for each $j=1,\ldots,n$, and $\ell=1,\ldots,n^{3}-n$.

For more details about the $GK$ curve, see [9].

3. The Weierstrass semigroup at certain $m+1$ points on $GK$ curve

In this section we will determine the Weierstrass semigroup $H(P_{\infty},P_{1},\ldots,P_{m})$, for $1\leq m\leq n$. To simplify the notation, we will denote $a=n^{2}-n+1$, $b=n^{3}$ and $c=n^{3}+1$. By the divisors of the rational functions $(x-a_{j})$, $y$ and $z$ given in (2), we have the following equivalences

[TABLE]

[TABLE]

[TABLE]

Let $1\leq m\leq n$ and let $1\leq k\leq a$, $0\leq i\leq n$ and $j_{s}\geq 0$ be integers such that

[TABLE]

So, the divisor

[TABLE]

is effective and using (3), (4) and (5) we have that

[TABLE]

The following lemma is important to show that a divisor is a discrepancy.

Lemma 3.1**.**

[7, Noether’s Reduction Lemma]** Let $D$ be a divisor, $P\in\mathcal{X}$ and let $K$ be a canonical divisor. If $\dim(\mathcal{L}(D))>0$ and $\dim(\mathcal{L}(K-D-P))\neq\dim(\mathcal{L}(K-D))$, then $\dim(\mathcal{L}(D+P))=\dim(\mathcal{L}(D))$.

Proposition 3.2**.**

The divisor $A^{\prime}$ is a discrepancy with respect to $P$ and $Q$ for any two distinct points $P,Q\in\{P_{\infty},P_{1},\ldots,P_{m}\}$.

Proof First, note that the equivalence of effective divisors in (7) gives a rational function $f\in\mathcal{L}(A^{\prime})$ with pole divisor equal to $A^{\prime}$. Thus, $\mathcal{L}(A^{\prime})\neq\mathcal{L}(A^{\prime}-P)$ for all $P\in\{P_{\infty},P_{1},\ldots,P_{m}\}$.

Now, we must prove that $\mathcal{L}(A^{\prime}-P)=\mathcal{L}(A^{\prime}-P-Q)$ and $\mathcal{L}(A^{\prime}-Q)=\mathcal{L}(A^{\prime}-Q-P)$, for all $P,Q\in\{P_{\infty},P_{1},\ldots,P_{m}\}$. By Lemma 3.1, it suffices to prove that $\mathcal{L}(K-A^{\prime}+P)\neq\mathcal{L}(K-A^{\prime}+P+Q)$, where $K$ is a canonical divisor. Taking $K=(n^{2}-2)cP_{\infty}$, we have that

[TABLE]

If $P_{\infty}\in\{P,Q\}$, without loss of generality, assume that $P=P_{\infty}$ and $Q=P_{1}$. Thus,

[TABLE]

and we have that

[TABLE]

So, $\mathcal{L}(A^{\prime}-Q)=\mathcal{L}(A^{\prime}-P-Q)$. Since $\mathcal{L}(A^{\prime})\neq\mathcal{L}(A^{\prime}-Q)=\mathcal{L}(A^{\prime}-P-Q)$ and $\mathcal{L}(A^{\prime})\neq\mathcal{L}(A^{\prime}-P)$, it follows that $\mathcal{L}(A^{\prime}-P)=\mathcal{L}(A^{\prime}-P-Q).$

If $P_{\infty}\not\in\{P,Q\}$, we can suppose that $P=P_{1}$ and $Q=P_{2}.$ In this case, we have that

[TABLE]

As above, it follows that $\mathcal{L}(A^{\prime}-Q)=\mathcal{L}(A^{\prime}-P-Q)$ and that $\mathcal{L}(A^{\prime}-P)=\mathcal{L}(A^{\prime}-P-Q).$

Therefore, $A^{\prime}$ is a discrepancy with respect to $P$ and $Q$ for any two distinct points $P,Q\in\{P_{\infty},P_{1},\ldots,P_{m}\}$. $\square$

Remark 3.3**.**

From (7) and Definition 2.5 follows that the divisor $((n^{2}-m-\sum_{s=1}^{m}j_{s})c-ina-kb)P_{\infty}+\sum_{s=1}^{m}(j_{s}c+ia+k)P_{s}$ is also a discrepancy with respect to $P$ and $Q$ for any two distinct points $P,Q\in\{P_{\infty},P_{1},\ldots,P_{m}\}$.

Theorem 3.4**.**

Let $a$, $b$, $c$, $P_{\infty},P_{1},\ldots,P_{m}$ be as above. For $1\leq m\leq n$, let

[TABLE]

Then, $\Gamma(P_{\infty},P_{1},\ldots,P_{m})=\Gamma_{m+1}$.

Proof By Proposition 3.2, $A^{\prime}$ is a discrepancy with respect to $P$ and $Q$ for any two distinct points $P,Q\in\{P_{\infty},P_{1},\ldots,P_{m}\}$. By Remark 3.3, the divisor $A=((n^{2}-m-\sum_{s=1}^{m}j_{s})c-ina-kb)P_{\infty}+\sum_{s=1}^{m}(j_{s}c+ia+k)P_{s}$ is a discrepancy with respect to $P$ and $Q$ for any two distinct points $P,Q\in\{P_{\infty},P_{1},\ldots,P_{m}\}$. Therefore, by Lemma 2.6, we have that $\Gamma_{m+1}\subseteq\Gamma(P_{\infty},P_{1},\ldots,P_{m})$.

Next, we show that $\Gamma(P_{\infty},P_{1},\ldots,P_{m})\subseteq\Gamma_{m+1}$. Let $\mathbf{n}=(n_{0},n_{1},\ldots,n_{m})\in\Gamma(P_{\infty},P_{1},\ldots,P_{m})$. By Definition 2.3 and Proposition 2.1, follows that $\mathbf{n}$ is minimal in $\nabla_{r}(\mathbf{n})$ for all $r$, $1\leq r\leq m+1$. By Lemma 2.4, $\mathbf{n}=(n_{0},n_{1},\ldots,n_{m})\in G(P_{\infty})\times G(P_{1})\times\cdots\times G(P_{m})$.

Note that $H(P_{s})=\langle an,b,c\rangle$, for all $1\leq s\leq m$, and then $n_{s}=j_{s}c+i_{s}a+k_{s}$, for some $j_{s}\geq 0$, $0\leq i_{s}\leq n$ and $1\leq k_{s}\leq a$. Let

[TABLE]

Then, $(f)_{\infty}=((n^{2}-m-\sum_{s=1}^{m}j_{s})c-ina-kb)P_{\infty}+(j_{1}c+ia+k)P_{1}+\cdots+(j_{m}c+ia+k)P_{m}$. Now, by Remark 3.3, $(f)_{\infty}$ is a discrepancy with respect to $P$ and $Q$ for any two distinct points $P,Q\in\{P_{\infty},P_{1},\ldots,P_{m}\}$ and so, by Lemma 2.6,

$\mathbf{f}=((n^{2}-m-\sum_{s=1}^{m}j_{s})c-ina-kb,j_{1}c+ia+k,\ldots,j_{m}c+ia+k)\in\Gamma(P_{\infty},P_{1},\ldots,P_{m})$.

Thus, $\mathbf{f}\in\nabla_{r}({\bf n})$, for some $1\leq r\leq m+1$, and, by Proposition 2.1, it follows that $\mathbf{f}$ is minimal in $\nabla_{r}(\mathbf{n})$ for all $r$, $1\leq r\leq m+1$. Therefore, by minimality of $\mathbf{f}$ and $\mathbf{n}$, we have that $\mathbf{f}=\mathbf{n}$ and $\Gamma(P_{\infty},P_{1},\ldots,P_{m})\subseteq\Gamma_{m+1}$.

$\square$

Example 3.5**.**

For $n=2$, we have $a=3,b=8,c=9$, and the curve $GK$ with affine equations

[TABLE]

In this case, the genus $g=10$ and, by Proposition 2.8, $H(P_{1})=H(P_{2})=H(P_{\infty})=\langle 6,8,9\rangle$ and then $G(P_{0})=G(P_{\infty})=\{1,2,3,4,5,7,10,11,13,19\}$. We have the following divisors

[TABLE]

where $\ell=1,2$. For this curve, taking $m=1$, by Theorem 3.4, we have that $\Gamma(P_{\infty},P_{1})=\{(9(3-j_{1})-6i-8k,9j_{1}+3i+k):0\leq i\leq 2,1\leq k\leq 3,j_{1}\geq 0\mbox{ and }9(3-j_{1})-6i-8k>0\}$, therefore

[TABLE]

Taking $m=2$, we have that

[TABLE]

[TABLE]

then

[TABLE]

Other different examples can be found explicitly in www.alonso.prof.ufu.br/Example.pdf.

4. Pure gaps and codes over the $GK$ curve

An element $(\alpha_{1},\ldots,\alpha_{m})\in G(P_{1},\ldots,P_{m})$ is called pure gap if $\dim(\mathcal{L}(\sum_{i=1}^{m}\alpha_{i}P_{i}))=\dim(\mathcal{L}(\sum_{i=1}^{m}\alpha_{i}P_{i}-P_{j}))$, for all $j=1,\ldots,m$. This concept was introduced by M. Homma and S.J. Kim in [11]. In [2], C. Carvalho and F. Torres used the concept of pure gaps to obtain codes whose minimum distance have bounds better than the Goppa bound.

Theorem 4.1**.**

[2, Theorem 3.3]** Let $Q_{1},\ldots,Q_{n},P_{1},\ldots,P_{m}$ be distinct $\mathbb{F}_{q}$-rational points of $\mathcal{X}$ and assume that $m\leq q$. Let $(\alpha_{1},\ldots,\alpha_{m}),(\beta_{1},\ldots,\beta_{m})\in\mathbb{N}_{0}^{m}$ and set $D=Q_{1}+\cdots+Q_{n}$ and $G=\sum_{i=1}^{m}(\alpha_{i}+\beta_{i}-1)P_{i}$. Let $d_{\Omega}$ be the minimum distance of the code $C_{\Omega}(D,G)$. If $(\alpha_{1},\ldots,\alpha_{m})$ and $(\beta_{1},\ldots,\beta_{m})$ are pure gaps at $P_{1},\ldots,P_{m}$, then $d_{\Omega}\geq deg(G)-(2g-2)+m$, where $g$ is the genus of $\mathcal{X}$.

Using the concept of discrepancy we have the following result to obtain pure gaps.

Proposition 4.2**.**

Let $A=\sum_{\ell=0}^{m}a_{\ell}P_{\ell}$, where $(a_{0},a_{1},\ldots,a_{m})\in\Gamma(P_{0},P_{1},\ldots,P_{m})$. Let $\ell\in\{0,1,\ldots,m\}$, if $\mathcal{L}(A-P_{\ell})=\mathcal{L}(A-2P_{\ell})$, then

$(a_{0},a_{1},\ldots,a_{{\ell}-1},a_{\ell}-1,a_{\ell+1},\ldots,a_{m})$* is a pure gap of $H(P_{0},P_{1},\ldots,P_{m})$.*

Proof.

In fact, by Lemma 2.6, the divisor $A$ is a discrepancy with respect to $P$ and $Q$ for any two distinct points $P,Q\in\{P_{0},P_{1},\ldots,P_{m}\}$. So, $\mathcal{L}(A-P_{\ell})=\mathcal{L}(A-P_{\ell}-Q)$ for any $Q\in\{P_{0},P_{1},\ldots,P_{m}\}\setminus\{P_{\ell}\}$. Thus, if $\mathcal{L}(A-P_{\ell})=\mathcal{L}(A-2P_{\ell})$, by definition of pure gap, follows that $(a_{0},a_{1},\ldots,a_{\ell-1},a_{\ell}-1,a_{\ell+1},\ldots,a_{m})$ is a pure gap of $H(P_{0},P_{1},\ldots,P_{m})$. ∎

For Corollary 4.3 and Lemma 4.4 in the following, consider the $GK$ curve over $\mathbb{F}_{n^{6}}$ with genus $g$. We remember that $a=n^{2}-n+1$, $b=n^{3}$, $c=n^{3}+1$ and $1\leq m\leq n$, and that $P_{\infty},P_{1},\ldots,P_{m}$ are the rational points given in Section 3.

Corollary 4.3**.**

If $2\leq k\leq a$, then $((n^{2}-m)c-kb,k,\ldots,k,k-1)$ is a pure gap of Weierstrass semigroup $H(P_{\infty},P_{1},\ldots,P_{m})$ on $GK$.

Proof.

In fact, first note that $((n^{2}-m)c-kb,k,\ldots,k,k)\in\Gamma(P_{\infty},P_{1},\ldots,P_{m})$ by taking $i=0$ and $j_{s}=0$, for all $s=1,\ldots,m$, in the Theorem 3.4. Let $A=((n^{2}-m)c-kb)P_{\infty}+\sum_{s=1}^{m}kP_{s}$. By the previous Lemma, we must prove that $\mathcal{L}(A-P_{m})=\mathcal{L}(A-2P_{m})$. Let $K=(2g-2)P_{\infty}=(n^{2}-2)cP_{\infty}$ be a canonical divisor. Note that $z^{k-2}(x-a_{1})\ldots(x-a_{m-1})\in\mathcal{L}(K-A+2P_{m})\setminus\mathcal{L}(K-A+P_{m})$ and so $\mathcal{L}(K-A+P_{m})\neq\mathcal{L}(K-A+2P_{m})$. Thus, by Lemma 3.1, it follows that $\mathcal{L}(A-P_{m})=\mathcal{L}(A-2P_{m})$. ∎

Proposition 4.4**.**

Let $\alpha<2g-1$ and $(\alpha,1,1,\ldots,1)\in G(P_{\infty},P_{1},\ldots,P_{m})$. If

i) $\exists\lambda,\beta,\gamma\in\mathbb{N}_{0}$, with $\lambda\geq m$, such that $\lambda c+\beta an+\gamma b=2g-1-\alpha$, or

ii) $2g-1-\alpha\geq(m-1)c$ and $\exists\beta,\gamma\in\mathbb{N}_{0}$ such that $\beta an+\gamma b=2g-1-\alpha$,

then $(\alpha,1,1,\ldots,1)$ is a pure gap.

Proof.

Let $\alpha<2g-1$ and $(\alpha,1,1,\ldots,1)\in G(P_{\infty},P_{1},\ldots,P_{m})$. Consider the divisor $A=\alpha P_{\infty}+P_{1}+\cdots+P_{m}$ and the canonical divisor $K=(2g-2)P_{\infty}$. We will see that, if the conditions i) or ii) as above are satisfied, then $\mathcal{L}(K-A)\neq\mathcal{L}(K-A+P_{\infty})$ and $\mathcal{L}(K-A)\neq\mathcal{L}(K-A+P_{i})$, for all $i=1,\ldots,m$. Thus, Lemma 3.1, we conclude that $(\alpha,1,1,\ldots,1)$ is a pure gap.

First, suppose that $\exists\lambda,\beta,\gamma\in\mathbb{N}_{0}$, with $\lambda\geq m$, such that $\lambda c+\beta an+\gamma b=2g-1-\alpha$. Since $\lambda\geq m$, we can write $\lambda=\lambda_{1}+\cdots+\lambda_{m}$, with $\lambda_{i}\geq 1$ for all $i=1,\ldots,m$. So, $(x-a_{1})^{\lambda_{1}}\ldots(x-a_{m})^{\lambda_{m}}y^{\beta}z^{\gamma}\in\mathcal{L}(K-A+P_{\infty})\setminus\mathcal{L}(K-A)$ and $\dfrac{(x-a_{1})\ldots(x-a_{m})}{x-a_{i}}\in\mathcal{L}(K-A+P_{i})\setminus\mathcal{L}(K-A)$, for all $i=1,\ldots,m$. Therefore, we have that $(\alpha,1,1,\ldots,1)$ is a pure gap.

Now, suppose that $2g-1-\alpha\geq(m-1)c$ and $\exists\beta,\gamma\in\mathbb{N}_{0}$ such that $\beta an+\gamma b=2g-1-\alpha$. So, $y^{\beta}z^{\gamma}\in\mathcal{L}(K-A+P_{\infty})\setminus\mathcal{L}(K-A)$ and $\dfrac{(x-a_{1})\ldots(x-a_{m})}{x-a_{i}}\in\mathcal{L}(K-A+P_{i})\setminus\mathcal{L}(K-A)$, for all $i=1,\ldots,m$. Therefore, we have that $(\alpha,1,1,\ldots,1)$ is a pure gap. ∎

Let us remember that given a code $C$ with parameters $[n,k,d]$, we define its information rate by $R=k/n$ and its relative minimum distance by $\delta=d/n$. These parameters allows us to compare codes with different length. In the following example we get a code that have better relative parameters than the corresponding one-point code given in [6, Table IV].

Example 4.5**.**

Consider the $GK$ curve over $\mathbb{F}_{3^{6}}$ with affine equations

[TABLE]

This curve has $6076$ $\mathbb{F}_{3^{6}}$-rational points and genus $g=99$. By Corollary 4.3 and Proposition 4.4, respectively, it follows that $(142,2,2,1)$ and $(155,1,1,1)$ are pure gaps at $P_{\infty},P_{1},P_{2},P_{3}$. By Theorem 4.1, the $4$-point code $C_{\Omega}(D,296P_{\infty}+2P_{1}+2P_{2}+P_{3})$ of dimension $k_{\Omega}=5869$ has minimum distance $d_{\Omega}\geq 109$.