Acute Semigroups, the Order Bound on the Minimum Distance and the Feng-Rao Improvements

Let ${\mathbb{N}}_{0}$ denote the set of all non-negative integers. A numerical semigroup is a subset $\Lambda$ of ${\mathbb{N}}_{0}$ containing $0$, closed under summation and with finite complement in ${\mathbb{N}}_{0}$. For a numerical semigroup $\Lambda$ define the genus of $\Lambda$ as the number $g=\#({\mathbb{N}}_{0}\setminus\Lambda)$ and the conductor of $\Lambda$ as the unique integer $c\in\Lambda$ such that $c-1\not\in\Lambda$ and $c+{\mathbb{N}}_{0}\subseteq\Lambda$. The elements in $\Lambda$ are called the non-gaps of $\Lambda$ while the elements in $\Lambda^{c}={\mathbb{N}}_{0}\setminus\Lambda$ are called the gaps of $\Lambda$. The enumeration of $\Lambda$ is the unique increasing bijective map $\lambda:{\mathbb{N}}_{0}\longrightarrow\Lambda$. We will use $\lambda_{i}$ for $\lambda(i)$. For further details on numerical semigroups we refer the reader to [12]. The first aim of this work is to give a new class of numerical semigroups containing the well-known classes of symmetric and pseudo-symmetric semigroups, Arf semigroups and semigroups generated by an interval and which in turn is not the whole set of numerical semigroups.

Let $F/{\mathbb{F}}$ be a function field and let $P$ be a rational point of $F/{\mathbb{F}}$. For a divisor $D$ of $F/{\mathbb{F}}$, let ${\mathcal{L}}(D)=\{0\}\cup\{f\in F^{*}\mid(f)+D\geqslant 0\}$. Define $A=\bigcup_{m\geqslant 0}{\mathcal{L}}(mP)$ and let $\Lambda=\{-v_{P}(f)\mid f\in A\setminus\{0\}\}=\{-v_{i}\mid i\in{\mathbb{N}}_{0}\}$ with $-v_{i}<-v_{i+1}$. It is well known that the number of elements in ${\mathbb{N}}_{0}$ which are not in $\Lambda$ is equal to the genus of the function field. Moreover, $v_{P}(1)=0$ and $v_{P}(fg)=v_{P}(f)+v_{P}(g)$ for all $f,g\in A$. Hence, $\Lambda$ is a numerical semigroup. It is called the Weierstrass semigroup at $P$.

Suppose $P_{1},\dots,P_{n}$ are pairwise distinct rational points of $F/{\mathbb{F}}_{q}$ which are different from $P$ and let $\varphi$ be the map $A\rightarrow{\mathbb{F}}_{q}^{n}$ such that $f\mapsto(f(P_{1}),\dots,f(P_{n}))$. For $m\geqslant 0$ the one-point Goppa code of order $m$ associated to $P$ and $P_{1},\dots,P_{n}$ is defined as $C_{m}=\varphi({\mathcal{L}}(\lambda_{m}P))^{\perp}$.

Suppose that $\lambda$ is the enumeration of $\Lambda$. For any $i\in{\mathbb{N}}_{0}$ let $N_{i}=\{j\in{\mathbb{N}}_{0}\mid\lambda_{i}-\lambda_{j}\in\Lambda\}$ and let $\nu_{i}=\#N_{i}$. The order bound on the minimum distance of the code $C_{m}$ is defined as

and it satisfies $d_{C_{m}}\geqslant d_{ORD}(C_{m})$ where $d_{C_{m}}$ is the minimum distance of the code $C_{m}$. A better bound is the refined order bound defined as

This one satisfies $d_{C_{m}}\geqslant d_{ORD}^{\varphi}(C_{m})\geqslant d_{ORD}(C_{m})$. The order bound depends only on the Weierstrass semigroup whereas the refined order bound depends on the Weierstrass semigroup and on the map $\varphi$ as well. These bounds can be found in [6, 13, 12]. We have that for all $i\geqslant\lambda^{-1}(2c-1)$, the sequence $(\nu_{i})$ is increasing and so, for all $i\geqslant\lambda^{-1}(2c-1)-1$, $d_{ORD}(C_{i})=\nu_{i+1}$. This equality makes it easier to compute the order bound, since we do not need to find a minimum in a set. The second goal of this work is to find an explicit formula for the smallest $m$ for which $d_{ORD}(C_{i})=\nu_{i+1}$ for all $i\geqslant m$. This is equivalent to search for the largest $i$ at which $(\nu_{i})$ is decreasing. We will give such a formula for the numerical semigroups in the class of acute semigroups.

On the other hand, given a designed minimum distance $\delta\in{\mathbb{N}}_{0}$, since $d_{ORD}(C_{m})=\min\{\nu_{i}\mid i>m\}$, the code $C_{m}$ with largest dimension among the codes $C_{i}$ with $d_{ORD}(C_{i})\geqslant\delta$ is with $m=m(\delta)=\max\{i\in{\mathbb{N}}_{0}\mid\nu_{i}<\delta\}.$ We call this code $C(\delta)$. Now, let ${\mathcal{F}}=\{f_{i}\in A\mid i\in{\mathbb{N}}_{0}\}$ be such that $v_{P}(f_{i})=v_{i}$. With this notation,

where $[u_{1},\dots,u_{n}]$ is the ${\mathbb{F}}_{q}$-vector space spanned by $u_{1},\dots,u_{n}$. Define

This is a code and it satisfies $\dim\widetilde{C}(\delta)\geqslant\dim C(\delta)$ because $\{i\in{\mathbb{N}}_{0}\mid\nu_{i}<\delta\}\subseteq\{i\in{\mathbb{N}}_{0}\mid i\leqslant m(\delta)\}$. Feng and Rao proved that the minimum distance of $\widetilde{C}(\delta)$ is larger than or equal to $\delta$ and hence, they are an improvement to one-point Goppa codes [7]. If moreover we take the morphism $\varphi$ into account we can drop the redundant rows in the parity check martix and define the code $\widetilde{C}_{\varphi}(\delta)=[\varphi(f_{i})\mid\nu_{i}<\delta,C_{i}\neq C_{i-1}]^{\perp}$. The third goal of this work is to characterize the numerical semigroups that satisfy

at least for one value of $\delta$. Notice that this is equivalent to ask for which numerical semigroups the sequence $(\nu_{i})$ has $\nu_{i}>\nu_{i+1}$ for at least one $i$.

This will give a characterization of a class of semigroups by means of a property on the sequence $(\nu_{i})$. We may ask how a numerical semigroup can be determined by its associated sequence $(\nu_{i})$. The last goal of this work is to prove that any numerical semigroup is uniquely determined by its sequence $(\nu_{i})$.

In Section II we give the definition of symmetric and pseudo-symmetric semigroup and some related known results [13, 12, 17], in Section III we define Arf numerical semigroups as in [4] and give some results and in Section IV we present numerical semigroups generated by an interval as in [9]. In Section V we introduce the definition of acute numerical semigroups and we prove that they include symmetric, pseudo-symmetric, Arf and interval-generated semigroups. We then find in Section VI, for acute numerical semigroups, the largest $m\in{\mathbb{N}}_{0}$ for which $\nu_{m}>\nu_{m+1}$ and hence the smallest $m$ for which $d_{ORD}(C_{i})=\nu_{i+1}$ for all $i\geqslant m$. We prove in Section VII that the only numerical semigroups for which $(\nu_{i})$ is always non-decreasing, that is, $d_{ORD}(C_{i})=\nu_{i+1}$ for all $i\in{\mathbb{N}}_{0}$, or equivalently $\{i\in{\mathbb{N}}_{0}\mid\nu_{i}<\delta\}=\{i\in{\mathbb{N}}_{0}\mid i\leqslant m(\delta)\}$ for all $\delta\in{\mathbb{N}}_{0}$, are ordinary numerical semigroups. Those are the numerical semigroups that are equal to $\{0\}\cup\{i\in{\mathbb{N}}_{0}\mid i\geqslant c\}$ for some non-negative integer $c$. We finally prove in Section VIII that a numerical semigroup can be uniquely determined by its sequence $(\nu_{i})$.

Symmetric numerical semigroups have been studied in [13, 12, 3].

Let us state now a proposition related to symmetric numerical semigroups.

The proof can be found in [13, Remark 4.2] and [12, Proposition 5.7]. It follows by counting the number of gaps and non-gaps smaller than the conductor and the fact that if $i$ is a non-gap then $c-1-i$ must be a gap because otherwise $c-1$ would also be a non-gap.

Notice that a symmetric numerical semigroup can not be pseudo-symmetric. Next proposition as well as its proof is analogous to Proposition II.3.

In [17] the authors prove that the set of irreducible semigroups, that is, the semigroups that can not be expressed as a proper intersection of two numerical semigroups, is the union of the set of symmetric semigroups and the set of pseudo-symmetric semigroups.

For further work on Arf numerical semigroups we refer the reader to [1, 18]. For results on Arf semigroups related to coding theory, see [2, 4].

Let us state now two results on Arf numerical semigroups that will be used later.

Let us give the definition of inductive numerical semigroups. They are an example of Arf numerical semigroups.

In order to show which are the only Arf pseudo-symmetric semigroups we need the following definition and lemma.

Now, $\lambda_{1}+c-1-l-\lambda_{1}=c-1-l\not\in\Lambda$ because otherwise $c-1\in\Lambda$. So $\lambda_{1}+c-1-l$ must belong to $Ap(\Lambda)$.

Now, if $\#Ap(\Lambda)=1$ then, by Remark III.10, $\lambda_{1}=1$ and $\Lambda={\mathbb{N}}_{0}$. But ${\mathbb{N}}_{0}$ is not pseudo-symmetric.

If $\#Ap(\Lambda)=2$ then, by Remark III.10, $\lambda_{1}=2$. But then $\Lambda$ must be hyperelliptic and so $\Lambda$ is not pseudo-symmetric.

So $\#Ap(\Lambda)$ must be $3$. Now Remark III.10 implies that $\lambda_{1}=3$ and that $1$ and $2$ are gaps. If $1=(c-1)/2$ then $c=3$ and this gives $\Lambda=\{0,3,4,5,6,\dots\}$. Else if $2=(c-1)/2$ then $c=5$ and this gives $\Lambda=\{0,3,5,6,7,\dots\}$. Finally, if $1\neq(c-1)/2$ and $2\neq(c-1)/2$, since $\Lambda$ is pseudo-symmetric, $c-2,c-3\in\Lambda$. But this contradicts Lemma III.3.

The next two propositions are two characterizations of Arf numerical semigroups.

On the other hand, if $\Lambda$ is such that $S(l)$ is a numerical semigroup for any $l\in\Lambda$ then, if $\lambda_{i}\geqslant\lambda_{j}\geqslant\lambda_{k}$ are in $\Lambda$, we will have $\lambda_{i}-\lambda_{k}\in S(\lambda_{k})$, $\lambda_{j}-\lambda_{k}\in S(\lambda_{k})$, $\lambda_{i}+\lambda_{j}-\lambda_{k}-\lambda_{k}\in S(\lambda_{k})$ and therefore $\lambda_{i}+\lambda_{j}-\lambda_{k}\in\Lambda$.

A numerical semigroup $\Lambda$ is generated by an interval $\{i,i+1,\dots,j\}$ with $i,j\in{\mathbb{N}}_{0}$, $i\leqslant j$ if

A study of semigroups generated by intervals was carried out by García-Sánchez and Rosales in [9].

This lemma is a reformulation of [9, Lemma 1].

Now, by definition of pseudo-symmetric semigroup, $(c-1)/2$ must be the first gap or the last gap of an interval of gaps. Suppose that it is the first gap of an interval of $n$ gaps. If it is equal to $1$ then $c=3$ and $\Lambda=\{0,3,4,5,6,\dots\}$. Otherwise $(c-1)/2>\lambda_{1}$. Then, if $\Lambda$ is pseudo-symmetric, the previous interval of non-gaps has length $n-1$. Since $\Lambda$ is generated by an interval, the first interval of non-gaps after $(c-1)/2$ must have length $n-1+j-i$ and since $\Lambda$ is pseudo-symmetric the interval of gaps before $(c-1)/2$ must have the same length. But since $\Lambda$ is generated by an interval, the interval of gaps previous to $(c-1)/2$ must have length $n+j-i$. This is a contradiction. The same argument proves that $(c-1)/2$ can not be the last gap of an interval of gaps. So, the only possibility for a pseudo-symmetric semigroup generated by an interval is when $(c-1)/2=1$, that is, when $\Lambda=\{0,3,4,5,6,\dots\}$.

Almost all rational points on a curve of genus $g$ over an algebraically closed field have Weierstrass semigroup of the form $\{0\}\cup\{i\in{\mathbb{N}}_{0}\mid i\geqslant g+1\}$. Such points are said to be ordinary. This is why we call these numerical semigroups ordinary [11, 5, 20]. Caution must be taken when the characteristic of the ground field is $p>0$, since there exist curves with infinitely many non-ordinary points [21].

Notice that ${\mathbb{N}}_{0}$ is an ordinary numerical semigroup. It will be called the trivial numerical semigroup.

It is well defined because of Remark V.4.

Roughly speaking, a numerical semigroup is acute if the last interval of gaps before the conductor is smaller than the previous interval of gaps.

In Figure 1 we summarize all the relations we have proved between acute semigroups, symmetric and pseudo-symmetric semigroups, Arf semigroups and semigroups generated by an interval.

In this section we will find a formula for the smallest $m$ for which $d_{ORD}(C_{i})=\nu_{i+1}$ for all $i\geqslant m$, for the case of acute semigroups. We will use the following well-known result on the values $\nu_{i}$.

Let $l=c-d-1$. Notice that $l$ is the number of gaps between the conductor and the dominant. Since $\Lambda$ is acute, the $l$ integers before $c^{\prime}$ are also gaps. Let us call $k=\lambda^{-1}(c^{\prime}+d)$. For all $1\leqslant i\leqslant l$, both $(c^{\prime}-i)$ and $(d+i)$ are in $D(k)$ because they are gaps and

Moreover, there are no more gaps in $D(k)$ because, if $j\leqslant c^{\prime}-l-1$ then $c^{\prime}+d-j\geqslant d+l+1=c$ and so $c^{\prime}+d-j\in\Lambda$. Therefore,

Now suppose that $j\geqslant k$. By Remark VI.2, $\lambda_{k}\geqslant c$ and so $\lambda_{j}=\lambda_{k}+j-k=c^{\prime}+d+j-k$. Then,

where

Notice that $A(j)\cap B(j)=\emptyset$ and hence

We have

So,