 # Pure gaps on curves with many rational places

We consider the algebraic curve defined by y^m = f(x) where m ≥ 2 and f(x) is a rational function over F_q. We extend the concept of pure gap to c-gap and obtain a criterion to decide when an s-tuple is a c-gap at s rational places on the curve. As an application, we obtain many families of pure gaps at two rational places on curves with many rational places.

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## 1. Introduction

Since Goppa introduced the Algebraic Geometric codes in the eighties , a lot of effort has been directed towards obtaining examples of codes with good parameters through different types of algebraic curves. It is well known that codes with optimal parameters are expected from algebraic curves with many rational places over finite fields. These are curves whose numbers of rational places are equal or close to the Hasse-Weil upper bound or other known bounds that may be specific to the curve. For a projective, absolutely irreducible, non-singular algebraic curve of genus over , the Hasse-Weil upper bound on the number of -rational places is When this quantity is attained, the curve is said to be maximal. Maximal curves only exist over .

In  and  Garcia, Kim and Lax exploit a local property at a rational place on an algebraic curve in order to improve the minimum distance of the code. Specifically, they show that the existence of consecutive gaps at a rational place can be used to increase the classical upper bound for the minimum distance by units. In  Homma and Kim obtain similar results using gaps at two rational places. They also define a special type of gap that they call pure gap and obtain further improvements. Then they use pure gaps to refine the parameters of codes constructed from the Hermitian curve over . In  Carvalho and Torres consider in detail gaps and pure gaps at more than two places. Since then, pure gaps have been exhaustively studied to show that they provide many other good codes; see [20, 29, 30, 23, 25, 5, 2]. The scope of these papers varies depending on the algebraic curve and the number of places considered.

The majority of maximal curves and curves with many rational places has a plane model of Kummer-type. For those curves with affine equation given by

 ym=f(x)λ where m≥2,λ≥1 and f(x) % is a separable polynomial over Fq,

general results on gaps and pure gaps can be found in [1, 5, 29, 18]. Applications to codes on particular curves such as the Giulietti-Korchmáros curve, the Garcia-Güneri-Stichtenoth curve, and quotients of the Hermitian curve can be found in [28, 19, 31]. In 

the authors use a decomposition of certain Riemann-Roch vector spaces due to Maharaj (Theorem

2.2) to describe gaps at one place arithmetically. The same idea has been used to investigate gaps and pure gaps at several places [2, 5, 18, 19, 29, 30, 31]. Here we continue exploring its capabilities by considering a different setting.

In this work we consider a Kummer-type curve defined by

 ym=f(x) where m≥2 and f(x) is a rational function % over Fq.

We extend the concept of pure gap to -gap and provide an arithmetical criterion to decide when an -tuple is a -gap at rational places. This result is then heavily used to obtain many families of pure gaps at two places on the curves obtained by Giulietti and Korchmáros in , Garcia and Quoos in , and Garcia, Güneri and Stichtenoth in . All these curves are known to have many rational places.

In the special case of two rational places, one can also determine the set of pure gaps using a method introduced by Homma and Kim in [17, Theorem 2.1]. For , let be a rational place and be the set of gaps at . There is a bijection from to given by

 β(n1):=min{n2:(n1,n2)∈H(P1,P2)},

which can be used to characterize the set of pure gaps at in the following way:

 (1) G0(P1,P2)={(n1,n2)∈G(P1)×G(P2):n1<β−1(n2) and β(n1)>n2}.

We use this approach to obtain additional families of pure gaps at two places on the Suzuki curve.

We conclude the paper with a summary of the parameters of codes constructed using our families of pure gaps.

## 2. Preliminary results

Let be a projective, absolutely irreducible, non-singular algebraic curve of genus defined over a finite field . Let be its function field over . For a function in , let and stand for its divisor and pole divisor, respectively. We denote by the set of places of , and by the free abelian group generated by the places of . The elements in are called divisors and can be written as

 D=∑P∈PFnPP with nP∈Z,% and nP=0 for almost all P∈PF.

The degree of a divisor is . The Riemann-Roch vector space associated to is defined by

 L(D):={z∈F:(z)≥−D}∪{0}.

We denote by the dimension of as a vector space over the field of constants .

Our convention is that and . For a rational place , that is, a place of degree one, the Weierstrass semigroup at is

 H(P):={n∈N:(z)∞=nP for some z∈F}∪{0}.

We say that is a non-gap at if , and a gap otherwise. Another way to characterize a gap is using the dimension of Riemann-Roch spaces, namely, is a gap at if and only if . As a consequence, the Weierstrass Gap Theorem asserts that there exist gaps at between and . There is a natural generalization of the notion of gap at distinct rational places of by setting

 H(P1,…,Ps):={(n1,…,ns)∈Ns0:(z)∞=n1P1+⋯+nsPs for some z∈F}

as the Weierstrass semigroup at . The elements in

 G(P1,…,Ps):=Ns0∖H(P1,…,Ps)

are called gaps, and there is a finite number of them. Gaps at several places can be described in terms of the dimensions of Riemann-Roch spaces: is a gap at if and only if

 ℓ(s∑i=1niPi−Pj)=ℓ(s∑i=1niPi) for some j∈{1,…,s}.

Homma and Kim  introduced the important concept of pure gap. An -tuple is a pure gap at if

 ℓ(s∑i=1niPi−Pj)=ℓ(s∑i=1niPi) for all j∈{1,…,s}.

The set of pure gaps at is denoted by . Clearly, is contained in . Carvalho and Torres [4, Lemma 2.5] showed that is a pure gap at if and only if .

Now we turn our attention to Kummer extensions. Let be the algebraic closure of and be the characteristic of .

###### Definition 2.1.

A Kummer extension is an algebraic function field defined by where is a rational function over with and For a zero or pole of , let be the place of associated to it. To be more precise, we assume the following.

1. are all places corresponding to zeros and poles of . We denote the multiplicity of by .

2. with are totally ramified places in the extension

3. For , let with be the places in lying over When is a totally ramified place in we denote simply by

4. The divisor of the function in is

 (y):=r∑i=1di∑j=1λidiPi,j=s∑i=1λiPi+r∑i=s+1di∑j=1λidiPi,j.
5. For a divisor of and any subfield , write . The restriction of to is

 D|E:=∑R∈PEmin{⌊nQe(Q|R)⌋:Q|R}R

where is the ramification index of over .

The following result by Maharaj is a key ingredient in our arithmetic characterization of -gaps at several places.

###### Theorem 2.2 ([22, Theorem 2.2]).

Let be a Kummer extension of degree defined by . Then, for any divisor of that is invariant under the action of , we have that

 L(D)=m−1⨁t=0L([D+(yt)]∣∣Fq(x))yt

where denotes the restriction of the divisor to .

Pure gaps are related to the improvement of the designed minimum distance of Algebraic Geometric codes. For the differential code associated to the divisors and , we denote its length, dimension and minimum distance by , and , respectively; see .

We recall that denotes a projective, absolutely irreducible, non-singular algebraic curve of genus .

###### Theorem 2.3 ([4, Theorem 3.4]).

Let be pairwise distinct -rational places on , and let , be pure gaps at with for each . Consider the divisors and . If every with for is a pure gap at , then

 d≥deg(G)−(2g−2)+s∑i=1(bi−ai+1).

## 3. c -gaps on Kummer extensions

We start this section by generalizing the concept of gap to -gap at several places.

###### Definition 3.1.

Let be pairwise distinct -rational places on . Given , we say that is a -gap at if

 ℓ(s∑i=1(ni−ci)Pi)=ℓ(s∑i=1niPi).

When , a -gap at is a pure gap at . Also, if is a -gap at then are all pure gaps at where .

Lundell and McCullough proved a bound for the minimum distance called generalized floor bound. Next we state their result using the -gap terminology.

###### Theorem 3.2 ([21, Theorem 3]).

Let be pairwise distinct -rational places on . Suppose that there exist , and with each such that and are -gaps at . Consider the divisors and . If then

 d≥deg(G)−(2g−2)+s∑i=1ci.

Theorem 2.3 can be seen as a particular case of Theorem 3.2.

There are instances in which we can apply Theorem 3.2, but not Theorem 2.3. This occurs when some and , are not pure gaps, or when and for some .

###### Theorem 3.3.

Let be pairwise distinct totally ramified places in the Kummer extension . Let . Then is a -gap at if and only if for every exactly one of the two following conditions is satisfied:

1. for all

###### Proof.

Let . Using the notation as in Definition 2.1, we have that

 s∑i=1niPi+(yt)=s∑i=1(ni+tλi)Pi+r∑i=s+1di∑j=1tλidiPi,j,

and the restriction of this divisor to is

 [s∑i=1niPi+(yt)]∣∣∣K(x)=s∑i=1⌊ni+tλim⌋Qi+r∑i=s+1⌊tλim⌋Qi.

By definition, is a -gap at if By Theorem 2.2, we have that

So

 ℓ(s∑i=1niPi) =m−1∑t=0ℓ(s∑i=1⌊ni+tλim⌋Qi+r∑i=s+1⌊tλim⌋Qi) and ℓ(s∑i=1(ni−ci)Pi) =m−1∑t=0ℓ(s∑i=1⌊ni−ci+tλim⌋Qi+r∑i=s+1⌊tλim⌋Qi).

We conclude that is a -pure gap at if and only if

 −ℓ(s∑i=1⌊ni−ci+tλim⌋Qi+r∑i=s+1⌊tλim⌋Qi)=0

for all . Since has genus , the latter identity holds if and only if for all either

 s∑i=1⌊ni+tλim⌋+r∑i=s+1⌊tλim⌋<0

or

 s∑i=1⌊ni+tλim⌋+r∑i=s+1⌊tλim⌋≥0 and ⌊ni+tλim⌋=⌊ni−ci+tλim⌋

for . This concludes the proof. ∎

Our next goal is to obtain families of pure gaps on specific curves using Theorem 3.3.

### 3.1. Pure gaps on the GK curve

The Giulietti-Korchmáros curve over is a non-singular curve in defined by the affine equations:

 GK:{Yq+1=Xq+X,Zq2−q+1=Yq2−Y.

It has genus , and the number of its -rational places is . The GK curve first appeared in  as a maximal curve over , since the latter number coincides with the Hasse-Weil bound, . The GK curve is the first example of a maximal curve that is not -covered by the Hermitian curve, provided that .

A plane model for the GK curve is given by the affine equation:

 yq3+1=(xq+x)((xq+x)q−1−1)q+1.

Let be its function field. We denote by the only place of associated to the pole of .

###### Proposition 3.4.

Let and be two totally ramified places different from on the GK curve. For and , let

 n1 :=1+α(q3+1)+β(q2−q+1) and n2 :=q5−3q3+2q2−q−1−α(q3+1)−β(q2−q+1).

Then the pair is a pure gap at .

###### Proof.

The places and are zeros of , so . For and , we have that if and only if . Furthermore,

 n1+t≡0(modq3+1)⟺{t=q3 if β=0t=(q2+1)(q−1) if β=1

and

 n2+t≡0(modq3+1)⟺{t=(q2+1)(q−1) if β=0t=q3 if β=1.

It remains to show that the condition Theorem 3.3(i) holds in the above cases. We compute

 ⌊n1+tq3+1⌋+⌊n2+tq3+1⌋+(q2−q)⌊tq2−q+1⌋+⌊−tq3q3+1⌋ = {q2−1+(q2−q)q−q3=−1 if t=q3q2−2+(q2−q)(q−1)−q3+q2−q+1=−1 if t=(q2+1)(q−1).

Therefore is a pure gap at . ∎

###### Proposition 3.5.

Let and be two totally ramified places different from on the GK curve. The pair is a -gap, but not a pure gap, at .

###### Proof.

We start by investigating when Theorem 3.3(ii) does not hold. For , we have that

 ⌊1+tq3+1⌋≠⌊tq3+1⌋

if and only if . In this case (i) becomes

 ⌊1+q3q3+1⌋+⌊q5−q3+q2−2q3+1⌋+(q2−q)⌊q3q2−q+1⌋+⌊−q6q3+1⌋ = 1+q2−2+(q2−q)q−q3=−1<0.

On the other hand, is a not a -gap since for and we have

 ⌊1q3+1⌋+⌊q5−2q3+q2−2q3+1⌋>0

and

 ⌊q5−2q3+q2−2q3+1⌋≠⌊q5−2q3+q2−3q3+1⌋.

###### Proposition 3.6.

Let and be two totally ramified places different from on the GK curve. For , let

 ni:=αi(q3+1)+βi(q2−q+1)+γi

where , and . Let

 δi,j:={1 if βi(q2−q+1)+γi≤βj(q2−q+1)+γj0 otherwise.

Then is a pure gap at if and only if

 q2+1−δ1,2−δ2,1−α1−α2>max{β1+γ1,β2+γ2}.
###### Proof.

By Theorem 3.3, is a pure gap at if and only if, whenever for some and , one has

 (2)

Clearly, if and only if . In particular, since , we get for . First, we note that if and only if . For , we obtain that

Thus, by (2), we have

 α1+δi,1+α2+δi,2+(q2−q)(q−βi)−ti<0,

which is equivalent to

 βi+γi

Hence, we obtain that is a pure gap at if and only if for . ∎

### 3.2. Pure gaps on curves with many rational places

We now consider curves with many rational places that appeared in . The first curve is defined over by

 X1:ym=(xqn−x)qn−1

where and . This curve has genus and rational places over .

###### Proposition 3.7.

Suppose is even. On the curve defined by

 yqn+1=(xqn−x)qn−1,

let be the unique place at infinity and be a totally ramified place different from . We have that:

1. is a -gap, but not a pure gap, at , and

2. is a pure gap at for .

###### Proof.

Let and . We have that and for . For and , we have

 ⌊n0+tλ0qn+1⌋≠⌊n0−1+tλ0qn+1⌋

if and only if , that is, . In this case,

 ⌊n0+tλ0qn+1⌋+⌊n1+tλ1qn+1⌋+(qn−1)⌊tλ1qn+1⌋=−q2n2+3(qn−2)2+(qn−1)qn−22=−2<0.

On the other hand, is not a -gap at since for and we have that

and

 ⌊n0+tλ0qn+1⌋+⌊n1+tλ1qn+1⌋+(qn−1)⌊tλiqn+1⌋=qn−2≥0.

We conclude that is not a pure gap.

Now suppose that and . As in the previous case, we have

 ⌊n0+tλ0qn+1⌋≠⌊n0−1+tλ0qn+1⌋

if and only if . Also,

 ⌊n1+tλiqn+1⌋≠⌊n1+tλi−1qn+1⌋

if and only if , that is, for , and for . In these cases

 = ⎧⎪⎨⎪⎩−q2n2+(qn−3)+qn−22+(qn−1)qn−22=−3 if t=(qn+2)/2−qn+1+qn−3+⌊α2⌋≤−1 if t=1

So is a pure gap. ∎

The second curve we consider in this section is defined over by

 X2:ym=(xq+1+x+1)qxq+1+xq+1

where a divisor of . The common roots of the numerator and the denominator belong to and satisfy . We denote by the .

Case 1: If then

 g=(q−1)(m−1)+m−d2 and N=q2m+d.

Case 2: If then

 g =(q−2)(m−1)+(m−d) and N ={(q2−1)m+2d if (q−1)/d≡0(mod3)(q2−1)m if (q−1)/d≢0(mod3).

Case 3: If then

 g=q(m−1) and N=(q2+1)m.

Let