CONES FROM MAXIMUM h-SCATTERED LINEAR SETS AND A STABILITY RESULT FOR CYLINDERS FROM HYPEROVALS SAM ADRIAENSEN JONATHAN MANNAERT PAOLO SANTONASTASO

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CONES FROM MAXIMUM h-SCATTERED LINEAR SETS AND A

STABILITY RESULT FOR CYLINDERS FROM HYPEROVALS

SAM ADRIAENSEN, JONATHAN MANNAERT, PAOLO SANTONASTASO,

AND FERDINANDO ZULLO

Abstract. This paper mainly focuses on cones whose basis is a maximum h-scattered lin-

ear set. We start by investigating the intersection sizes of such cones with the hyperplanes.

Then we analyze two constructions of point sets with few intersection sizes with the hy-

perplanes. In particular, the second one extends the construction of translation KM-arcs

in projective spaces, having as part at inﬁnity a cone with basis a maximum h-scattered

linear set. As an instance of the second construction we obtain cylinders with a hyperoval

as basis, which we call hypercylinders, for which we are able to provide a stability result.

The main motivation for these problems is related to the connections with both Hamming

and rank distance codes. Indeed, we are able to construct codes with few weights and to

provide a stability result for the codes associated with hypercylinders.

AMS subject classiﬁcation (2020): 51E20; 51E21; 94B05.

Keywords: linear set; scattered linear set; Hamming metric code; rank metric code.

1. Introduction

Scattered linear sets (and more generally scattered spaces) were deﬁned and investigated

for the ﬁrst time in 2000 by Blokhuis and Lavrauw in [4]. Since their introduction, scattered

linear sets have found fertile ground in Galois geometries and in coding theory, see e.g.

[19,25]. In this paper we will mainly focus our attention on maximum h-scattered linear sets.

Let Vbe an r-dimensional vector space over Fqnand let Λ = PG(V, Fqn) = PG(r−1, qn)

be the associated projective space. If Uis a k-dimensional Fq-subspace of V, then the set

of points

LU={huiFqn:u∈U\ {0}} ⊆ Λ

is said to be an Fq-linear set of rank k. An important notion related to linear sets is

the weight of a subspace Ω with respect to LU, which is a measure of how much of

the linear set is contained in Ω. If all the (h−1)-dimensional projective subspaces have

weight at most h, then LUis said to be h-scattered; see [11]. When h= 1 this exactly

coincides with the notion introduced by Blokhuis and Lavrauw in [4] and when h=r−1 it

Sam Adriaensen and Jonathan Mannaert, Department of Mathematics and Data Science, Vrije

Universiteit Brussel, Brussels, Belgium

Paolo Santonastaso and Ferdinando Zullo, Dipartimento di Matematica e Fisica, Universit`a degli

Studi della Campania “Luigi Vanvitelli”, Viale Lincoln, 5, I– 81100 Caserta, Italy

E-mail addresses:sam.adriaensen@vub.be, jonathan.mannaert@vub.be,

paolo.santonastaso@unicampania.it, ferdinando.zullo@unicampania.it.

arXiv:2210.09645v1 [math.CO] 18 Oct 2022

2 S. ADRIAENSEN, J. MANNAERT, P. SANTONASTASO, AND F. ZULLO

coincides with the notion introduced by Sheekey and Van de Voorde in [30]. The rank of an

h-scattered linear set in PG(r−1, qn) is bounded by rn/(h+1) and an h-scattered linear set

with this rank is called a properly maximum h-scattered linear set. For these linear

sets the intersection numbers with respect to the hyperplanes are known (see [4, 11, 33])

and interestingly they take exactly h+ 1 distinct values. In this paper we will ﬁrst study

cones having as basis a properly maximum h-scattered Fq-linear set in a complementary

space to the vertex. The possible intersection sizes of such a set with a hyperplane can

be easily derived from the intersection numbers of the basis with respect to hyperplanes.

Then we exploit two constructions of point sets in PG(r−1, qn) which arise from cones of

properly maximum h-scattered linear sets. More precisely, let LUbe a cone with basis a

properly maximum h-scattered linear set contained in a hyperplane π∞of PG(r, qn) and

let P=hviFqn∈PG(r, qn)\π∞. Then we can consider the following two point sets:

(1) B=LU0, where U0=U⊕ hviFq;

(2) K= (π∞\LU)∪(B \ π∞)=(π∞\ B)∪(B \ π∞).

The second construction can be seen as a generalization of the construction of translation

KM-arcs, which are point sets in PG(2,2n) of the projective plane that can be all obtained

by the second construction replacing the cone with a special type of linear set of rank n

on the line at inﬁnity (known as a club) with q= 2, see [12, Theorem 2.1]. For both of

the constructions we determine the possible intersection sizes with the hyperplanes, which

are strongly related to the intersection sizes with the hyperplanes of the chosen properly

maximum h-scattered linear set, obtaining sets with few intersection numbers with respect

to the hyperplanes; see [13]. As a special instance of Construction (2) we obtain the hyper-

cylinder, that is a cone with as basis a hyperoval, and as vertex a subspace of codimension

3, where the vertex is then deleted. We prove a stability result for hypercylinders obtaining

that when considering a point set with size close to the size of a hypercylinder and with

at most three possible intersection sizes with the hyperplanes, then it necessarily is a hy-

percylinder. The main tool regards some results on KM-arcs proved by Korchm´aros and

Mazzocca in [18] and two results of Calkin, Key and de Resmini in [8] on even sets.

The main motivation for studying these point sets is certainly related to coding theory.

Indeed, using the well-known correspondence between projective systems (or systems) and

Hamming metric codes (respectively rank metric codes), we are able to provide constructions

of codes with few weights in both Hamming and rank distances and to provide a stability

result for the codes associated with the hypercylinder (in the Hamming metric). The latter

codes deserve attention as they present only three nonzero weights.

The paper is organized as follows. In Section 2 we discuss some preliminaries that will

be useful later on. These focus mostly on linear sets, h-scattered linear sets, even sets and

KM-arcs. In Section 3 we prove some results on linear sets which are frequently used in

the paper and regard the size of certain families of linear sets. Section 4 is mainly devoted

ﬁrst to the study of cones with basis a properly maximum h-scattered linear set and then

to the determination of the intersection sizes of the hyperplanes with both Constructions

(1) and (2). As an instance of Construction (2) we obtain the hypercylinders. In Section

5 we provide a stability result for hypercylinders, making use of combinatorial techniques

and some combinatorial results on KM-arcs. In Section 6, after describing the connections

CONES FROM MAXIMUM h-SCATTERED LINEAR SETS AND A STABILITY RESULT 3

between Hamming/rank metric codes and projective systems/systems, we are able to con-

struct codes with few weights and to provide a stability result for those codes arising from

hypercylinders. This is indeed a consequence of the results obtained in the previous sec-

tions. Finally, we conclude the paper with Section 7 in which we summarize our results and

list some open problems/questions.

2. Preliminaries

We consider the projective space PG(r, q), with r≥2 and qa prime power, unless

otherwise stated.

Proposition 2.1 ( [17, Theorem 3.1.1]).The number of k-spaces in PG(r, q)containing a

ﬁxed h-space is

r−h

k−hq

=(qr−h−1)(qr−h−1−1) · · · (qr−k+1 −1)

(qk−h−1)(qk−h−1−1) · · · (q−1)

We will denote by [k+ 1]qthe number of points of a k-space of PG(r, q), i.e.

[k]q=qk−1

q−1.

In the paper we will frequently use the following notions.

Deﬁnition 2.2. Let Kbe a set of points in PG(r, q). Suppose that there exist spositive

integers m1<· · · < mssuch that for every k-space σ, with k≥1ﬁxed,

|σ∩ K| ∈ {m1, . . . , ms}.

Then we say that Kis of type {m1, . . . , ms}k. In case k= 1 and all the mi’s are even, K

is called an even set. If each of the integers mioccurs as the size of the intersection of K

with a k-space of PG(r, q), we say that Kis of type (m1, . . . , ms)k, and we call the mithe

intersection numbers.

We note that if Kis an even set in PG(r, q), then either qis odd and K ∈ {∅,PG(r, q)},

or qis even and Kintersects every subspace of dimension at least 1 in an even number of

points.

2.1. KM-arcs. Ovals and hyperovals are well studied objects in ﬁnite geometries.

Deﬁnition 2.3. Suppose that Ois a set of points in PG(2, q)such that no three points are

collinear. Then Ois called an oval of PG(2, q)if it has q+ 1 points, and a hyperoval if

it has q+ 2 points.

It can be seen that every line intersects an oval in 0, 1, or 2 points and every line intersects

a hyperoval in 0 or 2 points, and all of these cases occure. This makes an oval a set of type

(0,1,2)1and a hyperoval a set of type (0,2)1.

The standard example of an oval is a conic. Up to the action of PGL(3, q), there is

a unique conic, namely the solutions over F3

qto the equation Y2=XZ. Moreover some

classiﬁcation results for ovals are known. One of which was proven by Segre in 1955.

Theorem 2.4 ( [27, Theorem 1]).Suppose that qis odd. Then every oval in PG(2, q)is a

conic. Consequently, there are no hyperovals in PG(2, q).

4 S. ADRIAENSEN, J. MANNAERT, P. SANTONASTASO, AND F. ZULLO

However, for q > 4 even, there exist other examples of ovals; see e.g. [5]. Furthermore,

each oval in PG(2, q), qeven, can be extended to a hyperoval. Therefore, hyperovals always

exist in PG(2, q) with qeven.

Next, we deﬁne the KM-arcs, ﬁrst introduced and investigated by Korchm´aros and Maz-

zocca in [18].

Deﬁnition 2.5. AKM-arc of type tin PG(2, q)is a set of q+tpoints of PG(2, q)of

type (0,2, t)1.

It is immediately clear that KM-arcs are in fact generalizations of ovals and hyperovals,

by setting t= 1, respectively t= 2. However this generalization also has some interesting

properties.

Proposition 2.6 ( [18, Proposition 2.1]).Let Kbe a KM-arc of type tin PG(2, q), then:

•tdivides q;

•if 1< t < q, then qis even.

Recall that a KM-arc Kin PG(2, q) is called a translation KM-arc if there exists a line

`of PG(2, q) such that the group of elations with axis `and ﬁxing Kacts transitively on

the points of K \ `.

Finally, we list some known results on even sets in projective spaces, which have been

stated and proved using a coding theoretical approach. The ﬁrst regards a lower bound on

the size of an even set with respect to the lines, whereas the second one is a characterization

of those of minimum size.

Theorem 2.7 ( [8, Theorem 1]).Let Sbe an even set in PG(r, q), q even. Then |S| ≥

qr−1+ 2qr−2.

Deﬁnition 2.8. Let πand σbe complementary subspaces in PG(r, q), and take a set of

points S⊆σ. The cone Cwith vertex πand basis Sis the set of all points which lie on

a line intersecting both πand S, i.e.

C=[

P∈S

hP, πi.

We call C\πacylinder. If dim π=r−3, and Sis a hyperoval in σ, we call C\πa

hypercylinder.

Theorem 2.9 ( [8, Proposition 3]).For q≥4even, every point set in PG(3, q)of size

q2+ 2qof type (0,2, q)1is a hypercylinder.

Remark 2.10. The theorem above also holds for q= 2, because in that case, the complement

of such a set is a set of size [3]2of type (1,3)1, hence a hyperplane, and the complement of

a hyperplane is a hypercylinder.

2.2. Linear sets. Let Vbe an r-dimensional vector space over Fqnand let Λ = PG(V, Fqn) =

PG(r−1, qn). If Uis a k-dimensional Fq-subspace of V, then the set of points

LU={huiFqn:u∈U\ {0}} ⊆ Λ

CONES FROM MAXIMUM h-SCATTERED LINEAR SETS AND A STABILITY RESULT 5

is said to be an Fq-linear set of rank k. Note that when we use the notation LUfor an

Fq-linear set, we are formally considering both the set of points it deﬁnes and the underlying

subspace U.

If Ω = PG(W, Fqn) is a subspace of PG(r−1, qn), the intersection of LUwith Ω is the Fq-

linear set LU∩W. We say that Ω has weight iwith respect to LU, denoted as wLU(Ω) = i, if

the Fq-linear set LW∩Uhas rank i, i.e. wLU(Ω) = dimFq(U∩W). If Nidenotes the number

of points of Λ having weight i∈ {0, . . . , k}in LU, the following relations hold:

(1) |LU| ≤ [k]q,

(2) |LU|=N1+. . . +Nk,

(3) N1+N2(q+ 1) + . . . +Nk(qk−1+. . . +q+ 1) = [k]q.

Furthermore, LUis called scattered if it has the maximum number [k]qof points, or

equivalently, if all points of LUhave weight one.

Also, the following property concerning the weight of subspaces holds true.

Proposition 2.11 ( [24, Property 2.3]).Let LUbe an Fq-linear set of PG(r−1, qn)of rank

kand let Ωbe an s-space of PG(r−1, qn). Then Ω⊆LUif and only if wLU(Ω) ≥sn + 1.

We refer to [20] and [24] for comprehensive references on linear sets.

In [11], the authors introduced a special family of scattered linear sets, named h-scattered

linear sets; see [11, Deﬁnition 1.1].

Deﬁnition 2.12. Let hbe a positive integer such that 1≤h≤r−1. An Fq-linear set LUof

Λis called h-scattered (or scattered w.r.t. the (h−1)-dimensional subspaces) if hLUi= Λ

and each (h−1)-dimensional Fqn-subspace Ω = PG(W, Fqn)of Λhas weight in LUat most

The 1-scattered linear sets are the scattered linear sets generating Λ. The same deﬁnition

applied to h=rdescribes the canonical subgeometries of Λ (i.e. the copies of PG(r−1, q)

embedded in PG(r−1, qn)). If h=r−1 and dimFq(U) = n, then LUis h-scattered

exactly when LUis a scattered Fq-linear set with respect to the hyperplanes, introduced

in [30, Deﬁnition 14]; see also [21].

Theorem 2.13 bounds the rank of an h-scattered linear set.

Theorem 2.13 ( [11, Theorem 2.3]).If LUis an h-scattered Fq-linear set of rank kin

Λ = PG(r−1, qn), then one of the following holds:

•k=rand LUis a subgeometry PG(r−1, q)of Λ;

•k≤rn

h+1 .

An h-scattered Fq-linear set of maximum possible rank is said to be a maximum h-

scattered Fq-linear set. An h-scattered Fq-linear set of rank rn

h+1 is said to be a properly

maximum h-scattered Fq-linear set. Theorem 2.14 bounds the weight of the hyperplanes

with respect to a maximum h-scattered linear set.

Theorem 2.14 ( [11, Theorem 2.7]).If LUis a properly maximum h-scattered Fq-linear

set of Λ, then for any hyperplane σof Λwe have

h+ 1 −n≤wLU(σ)≤rn

h+ 1 −n+h.

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CONESFROMMAXIMUMh-SCATTEREDLINEARSETSANDASTABILITYRESULTFORCYLINDERSFROMHYPEROVALSSAMADRIAENSEN,JONATHANMANNAERT,PAOLOSANTONASTASO,ANDFERDINANDOZULLOAbstract.Thispapermainlyfocusesonconeswhosebasisisamaximumh-scatteredlin-earset.Westartbyinvestigatingtheintersectionsizesofsuchconeswiththehyperplanes...

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CONES FROM MAXIMUM h-SCATTERED LINEAR SETS AND A STABILITY RESULT FOR CYLINDERS FROM HYPEROVALS SAM ADRIAENSEN JONATHAN MANNAERT PAOLO SANTONASTASO

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