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Constructing saturating sets in projective spaces using

subgeometries

Lins Denaux

Ghent University

Abstract

A -saturating set of PG(N, q) is a point set S such that any point of PG(N, q) lies ina subspace of dimension at most spanned by points of S. It is generally known that a

-saturating set of PG(N, q) has size at least c · qN−

+1 , with c > 13 a constant.

Our main result is the discovery of a -saturating set of size roughly (+1)(+2)2 q

N−

+1

if q = (q′)+1, with q′ an arbitrary prime power. The existence of such a set improvesmost known upper bounds on the smallest possible size of -saturating sets if < 2N−1

3 .As saturating sets have a one-to-one correspondence to linear covering codes, this resultimproves existing upper bounds on the length and covering density of such codes.

To prove that this construction is a -saturating set, we observe that the affine partsof q′-subgeometries of PG(N, q) having a hyperplane in common, behave as certain lines ofAG(+ 1, (q′)N

). More precisely, these affine lines are the lines of the linear representation

of a q′-subgeometry PG(, q′) embedded in PG(+ 1, (q′)N

).

Keywords: Affine spaces, Covering codes, Linear representations, Projective spaces, Saturatingsets, Subgeometries.Mathematics Subject Classification: 05B25, 94B05, 51E20.

1 Motivation

The main topic of this article are -saturating sets of the Desarguesian projective space PG(N, q).A -saturating set of PG(N, q) is a point set S such that any point of PG(N, q) lies in asubspace of dimension at most spanned by points of S. These combinatorial structures are veryinteresting from a coding-theoretical point of view, since they have a one-to-one correspondenceto linear covering codes with covering radius +1 (see Subsection 4.1). The existence of a small-saturating set implies the existence of a ( + 1)-covering code with small length, a propertywhich is generally desired.

2 Preliminaries

Throughout this work, we assume N ∈ N \ {0} and ∈ {0, 1, . . . , N}. Whenever we shiftour geometrical perspective to its coding theoretical counterpart, we assume r ∈ N \ {0, 1}and R ∈ {1, . . . , r} (see Subsection 4.1). Furthermore, we assume q and q′ to be arbitraryprime powers. For the purpose of this article, it is useful to keep in mind that the assumptionq = (q′)+1 (equivalently, q = (q′)R) will often be made.

We will denote the Galois field GF(q) of order q by Fq and the Desarguesian projective spaceof (projective) dimension N over Fq by PG(N, q). By omitting a hyperplane in PG(N, q), wenaturally obtain the Desarguesian affine space of dimension N over Fq, which we will denote byAG(N, q). Furthermore, define the value

θN :=qN+1 − 1

q − 1,

1

which equals the number of points in PG(N, q).

Definition 2.1. Let m ∈ N\{0}. A frame of the projective geometry PG(m, q) is a set of m+2points of which no m+ 1 points are contained in a hyperplane.

The notion of a q′-subgeometry will be a key concept throughout this article. For a detaileddescription on subgeometries, see [22, p. 103].

Definition 2.2. Let m,m′ ∈ N, m′ 6 m, and suppose q = (q′)+1. An m′-dimensional q′-subgeometry B of PG(m, q) is a set of subspaces (points, lines, . . . , (m′ − 1)-dimensional sub-spaces) of PG(m, q), together with the incidence relation inherited from PG(m, q), such that Bis isomorphic to PG(m′, q′).If m′ = 1 or m′ = 2, we will often call B a q′-subline or a q′-subplane of PG(m, q), respectively.Moreover, we will denote the m′-dimensional subspace of PG(m, q) spanned by the points of Bby 〈B〉q. If m′ = m, we will omit the dimension and simply call B a q′-subgeometry of PG(m, q).Lastly, whenever q′ is clear from context, the prefix ‘q′-’ will often be omitted.

If = 1, a q′-subgeometry is obviously better known under the name Baer subgeometry, by farthe most studied subgeometry of projective spaces. One can find a short survey on Baer sublinesand Baer subplanes in [5] and on general Baer subgeometries in [8].Although a lot is known about q′-subgeometries, we will only use a few properties concerningthese structures, one of which is the following. The proof is done by considering the underlyingvector space of the projective geometry.

Lemma 2.3 ([5, Theorems 2.6 and 2.8] and [8, Lemma 1]). Let m ∈ N\{0} and let q be square.For each frame of PG(m, q), there exists a unique Baer subgeometry containing each point ofthe frame.

Using the exact same arguments as used in [5, 8], one can easily generalise the proof of thelemma above to arbitrary subgeometries:

Lemma 2.4. Let m ∈ N \ {0} and suppose q = (q′)+1. For each frame of PG(m, q), thereexists a unique q′-subgeometry containing each point of the frame.

Basically, every choice of frame in PG(m, q) and subfield in Fq results in finding a uniquesubgeometry defined over that subfield and containing the frame.

From this, we can deduce the following property.

Lemma 2.5. Let m ∈ N \ {0} and suppose q = (q′)+1. Consider an (m − 1)-dimensionalq′-subgeometry C of PG(m, q) and define Σ := 〈C〉q. Let L be a q′-subline of PG(m, q) having apoint in common with C and spanning a line ℓ * Σ. Then there exists a unique q′-subgeometrycontaining both C and L.

Proof. Let P be the unique point in C ∩ L. Consider a frame FC := {P,P1, P2, . . . , Pm} ofC. Naturally, FC is a frame of Σ as well, uniquely determining, by Lemma 2.4, the (m − 1)-dimensional q′-subgeometry C. For any two distinct points Q1, Q2 of L \ {P}, the set F :={P1, P2, . . . , Pm}∪{Q1, Q2} is a frame of PG(m, q). Hence, by Lemma 2.4, there exists a uniquem-dimensional q′-subgeometry B containing each point of F .Note that the set {P1, P2, . . . , Pm} spans an (m − 1)-dimensional q′-subgeometry C′ ⊆ B andthat the set {Q1, Q2} spans a q′-subline L′ ⊆ B. As C′ (spanning Σ) plays the role of an (m−1)-dimensional subspace of B, and L′ (spanning ℓ) plays the role of a line of B, by Grassmann’sidentity, these two objects have a point of B in common, necessarily equal to Σ ∩ ℓ = P . Thismeans that FC ⊆ C′ and thus, by Lemma 2.4, C = C′. The same holds for L′ and L, as bothcontain the frame {P,Q1, Q2}.

2

Now we introduce the most important definition of this work.

Definition 2.6. Let S be a point set of PG(N, q).

1. A point P ∈ PG(N, q) is said to be -saturated by S (or, conversely, the set S -saturatesP ) if there exists a subspace through P of dimension at most that is spanned by pointsof S. If is clear from context, the prefix ‘−’ is often omitted.

2. The set S is a -saturating set of PG(N, q) if is the least integer such that all points ofPG(N, q) are -saturated by S.

As reasoned in Section 1 (see also Subsection 4.1), it is justifiable to study small -saturating setsof PG(N, q), as these objects give rise to covering codes with good properties. In light of this,we will adopt the following notation, which is widely used in the literature (e.g. [2, 3, 13, 14]).

Notation 2.7. sq(N, ) := min {|S| : S is a -saturating set of PG(N, q)}.

The main research concerning saturating sets focuses on finding small upper bounds on sq(N, ).

3 Outline and main results

General preliminaries can be found in Section 2. Section 4 formalises the correspondence betweensaturating sets and covering codes. In light of this, relevant definitions and notation within acoding theoretical context are introduced. Furthermore, we state a quasi-trivial lower bound onthe size of a -saturating set, which naturally gives rise to the main research goal of this topic(see Open Problem 5.0.1). The core of this work can be found in Section 7, although Section 6in itself presents an interesting, stand-alone result.

Section 6 describes an isomorphism between two point-line geometries. One is the linear rep-resentation T ∗(D,m,q′) of a subgeometry D,m,q′

∼= PG(, q′) ⊆ PG( + 1, (q′)m

). The other is

a newly introduced point-line geometry Y (,m, q′), embedded in PG(m, (q′)+1

), of which the

lines are affine parts of subgeometries isomorphic to PG(m, q′) (see Definition 6.1.1).

Theorem 6.2.4. Let m ∈ N \ {0}. Then the point-line geometries Y (,m, q′) and T ∗(D,m,q′)are isomorphic.

Consequently, one can transfer natural notions of parallelism and independence of concurrentlines from T ∗(D,m,q′) to Y (,m, q′) (see Subsection 6.3). As a side note, we make the readeraware of the existence of an explicit isomorphism between this newly introduced point-linegeometry Y (,m, q′) and the point-line geometry X(,m, q′) introduced by De Winter, Rotteyand Van de Voorde [18] (see Subsection 6.4).

Section 7 discusses the main result of this work by presenting a general upper bound on sq(N, ),q = (q′)+1. This is obtained by constructing a -saturating set of PG(N, q) as a mix of severaldistinct, partially overlapping q′-subgeometries. The technique used to prove the saturationproperty of this construction relies on the results obtained in Section 6.Although a precise and extensive upper bound is given by Theorem 7.2.9, we present the followingconsequence as our main result, which is slightly weaker but far easier to comprehend.

Theorem 7.2.10. Let 1 < < N and let q = (q′)+1 for any prime power q′. Then

sq(N, ) 6(+ 1)( + 2)

2(q′)N− + (+ 1)

(q′)N− − 1

q′ − 1.

Translating the result above in coding theoretical terminology (see Subsection 4.1), one obtainsthe following.

3

Corollary 7.2.11. Let 2 < R < r and let q = (q′)R for any prime power q′. Then

ℓq(r,R) 6R(R+ 1)

2(q′)r−R + (R− 1)R

(q′)r−R − 1

q′ − 1.

For any infinite family of covering codes of length equal to the upper bound above, the followingholds for its asymptotic covering density:

µq(R) <

((R− 1)R

)R

R!

(1 +

1

q′+ · · ·+ 1

(q′)R−1

)R

<

(e(R − 1)

q − 1

q − (q′)R−1

)R

,

with e ≈ 2.718... being Euler’s number.

Plenty of extensive research has already been done concerning the topic of saturating sets andcovering codes. Therefore, Section 5 provides a careful comparison between our main resultsand relevant known results from the literature.

4 Covering codes and the research goal

Based on the way to approach this topic of research, the literature is divided. On the one hand,one can observe the topic geometrically by analysing small -saturating sets of PG(N, q). Onthe other hand, one can convert this geometrical point of view to a coding theoretical one byinvestigating covering codes of small length.

4.1 Translation to covering codes

In this subsection, we aim to formalise the correspondence between saturating sets and coveringcodes described in Section 1.

Definition 4.1.1. A q-ary linear code of length n and codimension (redundancy) r is said tohave covering radius R if R is the least integer such that every vector of Fn

q lies within Hammingdistance1 R of a codeword. Such a code will be called an [n, n− r]qR code.

Whenever linear codes are investigated with the goal of optimising the length or (co)dimensionwith respect to the covering radius, such codes are often called covering codes. These type ofq-ary linear codes have a wide range of applications; for a description of several examples ofsuch applications, see [12, Section 1].Suppose that S is a point set of PG(r− 1, q) of size n and let H be a q-ary (r× n)-matrix withthe homogeneous coordinates of the points of S as columns. Then S is an (R−1)-saturating setof PG(r − 1, q) if and only if H is a parity check matrix of an [n, n− r]qR code. This describesa one-to-one correspondence between saturating sets of projective spaces and linear coveringcodes. More specifically, any -saturating set S of PG(N, q) corresponds to an [n, n− r]qR codewith

n = |S|, r = N + 1 and R = + 1.

Due to this correspondence, the problem of finding small -saturating sets in PG(N, q) can betranslated to finding [n, n − r]qR codes of small length. In light of this, we adopt the notationof the length function ℓq(r,R).

Definition 4.1.2 ([7, 9]). The length function ℓq(r,R) is the smallest length of a q-ary linearcode with covering radius R and codimension r.

1The Hamming distance between two vectors of Fnq equals the number of positions in which they differ.

4

Note thatℓq(r,R) = sq(r − 1, R − 1).

From a coding theoretical perspective, it is interesting to analyse the extent to which the spheresof radius R centered at the codewords of an [n, n − r]qR code C overlap. This is done byinvestigating the covering density µq(n, r, C) of the code C, which equals the ratio of the totalvolume of these qn−r spheres to the volume of the space Fn

q (see e.g. [2, 3, 4, 10, 11, 12]):

µq(n, r, C) :=1

qn

(qn−r

R∑

i=0

(q − 1)i(n

i

))=

1

qr

R∑

i=0

(q − 1)i(n

i

)> 1.

Note that the latter inequality is sharp if and only if C is a perfect code.Analogous to the work of Davydov et al. [12], for a given R ∈ {1, . . . , n} and fixed prime powerq, we will call an infinite sequence of q-ary linear [n, n − rn]qR codes Cn an infinite family ofcovering codes, and denote such a family with AR,q. Given such an infinite family of coveringcodes AR,q, the asymptotic behaviour of the covering density when n tends to infinity is a topicof investigation. In light of this, we define its asymptotic covering densities

• µq(R,AR,q) := lim infn→∞

µq(n,R, Cn), and

• µ∗q(R,AR,q) := lim sup

n→∞µq(n,R, Cn).

If the infinite family of covering codesAR,q is clear from context, we will write µq(R) (respectivelyµ∗q(R)) instead of µq(R,AR,q) (respectively µ∗

q(R,AR,q)).

As the authors of [12] point out, given an infinite family AR,q for which rn+1 < rn for some n,one can replace Cn+1 by any 1-extension of Cn to obtain a code with a better covering density.Hence one may assume that the sequence of codimensions rn of AR,q is non-decreasing. In lightof this, a code Cn is called a supporting code of AR,q if rn > rn−1 (and a filling code otherwise).Davydov et al. [12] introduce and solve two open problems. The first open problem concernsthe search for an infinite family of covering codes AR,q for which µ∗

q(R,AR,q) = O(q); in thiscase, the corresponding family AR,q is said to be optimal [12, Open Problem 1]. The authorsnote that, in order to solve this open problem, it suffices to find a solution to the following:

Open Problem 4.1.3 ([12, Open Problem 2]). For any covering radius R > 2, construct R

infinite families of covering codes A(0)R,q,A

(1)R,q, . . . ,A

(R−1)R,q such that for each γ = 0, 1, . . . , R − 1

the supporting codes of A(γ)R,q are [nu, nu−ru]qR codes with codimension ru = Ru+γ and length

nu = f(γ)q (ru) with f

(γ)q (r) = O

(q

r−RR

)for any u > u0, with u0 a constant which may depend

on the family.

The authors of [12] managed to solve this open problem for arbitrary covering radius R > 2 andq = (q′)R (see [12, Section 4] and Subsection 5.2). We managed to do the same, in a somewhatmore effective way, as we could take u0 = 1 independent of the infinite family of covering codes,

and in most cases found a substantially smaller polynomial function f(γ)q (r) in q of which the

leading coefficient is quadratic in R. As a consequence, the infinite families of covering codes weobtained have a generally improved asymptotic covering density (see Subsection 5.2).

4.2 A lower bound on sq(N,) and ℓq(r,R)

In order to know which saturating sets are viewed as being ‘small’, we will be guided by thefollowing lower bound on the size of arbitrary -saturating sets. Several variants of this boundwere already known in the literature [2, 3, 4, 12, 13, 14], but some only state the bound forspecific values of , while others describe an approximate lower bound for large values of q.

5

Proposition 4.2.1. Let S be a -saturating set of PG(N, q), 6 N . Then

|S| > + 1

e· q

N−

+1 +

2,

with e ≈ 2.718... being Euler’s number.

Proof. Note that |S| > +1. Indeed, if this would not be the case, all points of S would span asubspace of dimension at most − 1 < N . In such a situation, it is impossible for S to saturateall points of the N -dimensional projective geometry PG(N, q), a contradiction.Hence, we can consider the set Π6 of all subspaces spanned by +1 distinct points of S; thesesubspaces are each of dimension at most . As S saturates PG(N, q), we know that Π6 has tocover the latter, thus ( |S|

+ 1

)θ > θN .

Expanding the binomial above and rearranging the inequality, we get

|S| (|S| − 1) (|S| − 2) · · · (|S| − ) > (+ 1)! · θNθ

> (+ 1)! · qN−, (1)

the last inequality being valid if and only if 6 N .

Note that the map f : N \ {0} → R : N 7→ N√N !

Nis strictly decreasing, with lim

N→∞f(N) = 1

e.

Therefore, we know that f(N) > 1efor all N ∈ N \ {0}, or, equivalently, N

√N ! > N

e. Combining

this with (1), after taking the (+ 1)th root of the left- and right-hand side, we obtain

+1√

|S| (|S| − 1) (|S| − 2) · · · (|S| − ) >+ 1

e· q

N−

+1 .

Applying the AM-GM inequality to the left-hand side finishes the proof.

Roughly speaking, Proposition 4.2.1 implies that

sq(N, ) > c · qN−

+1 , or, equivalently, ℓq(r,R) > c · Rqr−RR , (2)

for any N , and q (equivalently, for any r, R and q), where c > 13 is a constant independent

of these parameters. Naturally, researchers aim to prove that (2) is sharp by constructing small-saturating sets of PG(N, q) or, equivalently, constructing [n, n− r]qR covering codes of smalllength. This gives rise to Open Problem 5.0.1.

5 Comparison with relevant known results

In Subsection 4.2, we deduced a general lower bound on the size of a saturating set (equivalently,on the length of a covering code). This gives rise to the following.

Open Problem 5.0.1. Find a value c > 0, independent of q (and preferably independent of Nand as well), such that

sq(N, ) 6 c · qN−

+1 ,

or, equivalently, find a value c > 0, independent of q (and preferably independent of r and R aswell), such that

ℓq(r,R) 6 c · Rqr−RR .

With the exception of Remark 5.0.2, all mentioned results within this section solve the openproblem above for specific values of N , and q (equivalently, r, R and q), some in a moreeffective way than others. To give an overview, Open Problem 5.0.1 is solved if

6

1. N + 1 ≡ 0 (mod + 1) (equivalently, r ≡ 0 (mod R)); see Subsection 5.1.

2. q = (q′)+1 (equivalently, q = (q′)R); see Subsection 5.2.

3. N + 1 ≡ s +1′+1 (mod + 1) and q = (q′)

′+1, with ′ + 1 | + 1 and s ∈ {1, 2, . . . , ′}(equivalently, r ≡ s R

R′ (mod R) and q = (q′)R′

, with R′ | R and s ∈ {1, 2, . . . , R′− 1}); seeRemark 5.2.6.

Remark 5.0.2. Some results present upper bounds that are slightly larger than the desiredone described in Open Problem 5.0.1. More specifically, the authors of articles [2, 3, 4, 13, 15],some with the aid of computer searches, present upper bounds on sq(N, ), ∈ {1, 2}, of thefollowing form:

sq(N, ) 6 c · qN−

+1 +1√

ln q,

with c > 0 a constant independent of N , and +1√ln q a relatively small factor dependent on q.

One can immediately see that, if q = (q′)+1 = (q′)R, our main results (Theorem 7.2.10 andCorollary 7.2.11) also solve Open Problem 5.0.1 for c independent of q and N (equivalently,independent of q and r), and linearly dependent on (equivalently, linearly dependent on R).In this section, we will carefully compare these new results to relevant known results fromthe literature. Depending on the setting of each of these relevant results, we will make thecomparison from a geometrical point of view (Theorem 7.2.10) or a coding theoretical point ofview (Corollary 7.2.11).

5.1 Known results with assumptions on N and (correspondingly r and R)

A simple, recursive upper bound on sq(N, ) can be obtained geometrically by observing satu-rating sets in two disjoint subspaces spanning the ambient geometry. As stated in [16, Theorem5], the same bound arises from the direct sum construction of linear codes over a common finitefield.

Result 5.1.1 ([24, Lemma 10]). sq(N1 +N2 + 1, 1 + 2 + 1) 6 sq(N1, 1) + sq(N2, 2).

Corollary 5.1.2. Suppose that N + 1 is a multiple of + 1. Then sq(N, ) 6 ( + 1)θk, with

k := N−+1 .

Proof. By induction on . If = 0, this is a trivial statement. Inductively using Result 5.1.1,we obtain

sq(N, ) 6 sq(N − k − 1, − 1) + sq(k, 0) 6 θk + θk.

As shown in the proof of Theorem 7.2.9, the upper bound of Theorem 7.2.9 (and hence the oneof Theorem 7.2.10 as well) does not improve the upper bound above if N + 1 is a multiple of+ 1.Although, for N + 1 a multiple of + 1, Corollary 5.1.2 already solves Open Problem 5.0.1 (forc independent of N and ), we want to stress that better upper bounds concerning this specialcase are known in the literature. Davydov [10, Theorem 5.1] and Davydov and Ostergard[16, Theorem 7] slightly improved the bound above in case k = 1 and = 1, 2, respectively.The constructions behind these results are commonly denoted as the ‘oval plus line’ and ‘twoovals plus line’ constructions; in [12, Theorems 6.1 and 6.2], these bounds are generalised.Davydov, Marcugini and Pambianco [14, Theorem 1] managed to generalise this ‘oval(s) plusline’ construction to a -saturating set in PG(2 + 1, q). Using a coding-theoretical tool called‘qm-concatenating constructions’ [10, 11, 12, 14], they generalised their results even further andimproved the upper bound depicted in Corollary 5.1.2 under some minor restrictions on theparameters.

Furthermore, the same authors obtained the following results for the case R even, r ≡ R2

(mod R) and compared them with [12, Corollary 7.2].

7

Result 5.1.3 ([14, Theorem 2]). Let R > 2 be even. Let p be prime, q = p2η, η > 2, r = tR+ R2 ,

t > 1. The following constructive upper bounds on the length function hold:

1. ℓq(r,R) 6 R(1 +

√q−1√

q(φ(√q)−1)

)q

r−RR +R

⌊q

r−2RR

− 12

⌋+ R

2 fq(r,R) if p > 3, and

2. ℓq(r,R) 6 R(1 + 1

p+ 1√

q

)q

r−RR +R

⌊q

r−2RR

− 12

⌋+ R

2 fq(r,R) if p > 7,

where φ(q) is the order of the largest proper subfield of Fq and

fq(r,R) :=

{0 if t /∈ {4, 6},(q + 1)q

r−4RR

− 12 if t ∈ {4, 6}.

One can check that Corollary 7.2.11 does not improve this result, nor [12, Corollary 7.2], forgiven constraints on r, R and q.

5.2 Known results assuming q = (q′)+1 = (q′)R

In this subsection, we discuss some relevant known results based on the assumption that q =(q′)+1 (equivalently, q = (q′)R). This assumption allows mathematicians to exploit the useof q′-subgeometries. In the literature, one can notice two main approaches for constructingsaturating sets using subgeometries; we will call these two approaches the strong blocking setapproach and the mixed subgeometry approach.

The strong blocking set approach

The strong blocking set approach is based on constructing strong blocking sets in PG(N, q′).A ( + 1)-fold strong blocking set of PG(N, q′) is a point set that meets any -dimensionalsubspace in a set of points spanning said subspace. Although ( + 1)-fold strong blocking setswere introduced in [12, Definition 3.1], these are also known as generator sets ([19, Definition2]) or cutting blocking sets ([6, Definition 3.4]) in case = N − 1.Strong blocking sets directly generate saturating sets, as one can prove that (+ 1)-fold strongblocking sets are -saturating sets of the ambient geometry PG

(N, (q′)+1

)[12, Theorem 3.2].

This strong blocking set approach led to several results solving Open Problem 5.0.1, and are oftengeneralised using qm-concatenating constructions. For example, the following results considerthe case r 6≡ 0 (mod R) for R = 3 (equivalently, N + 1 6≡ 0 (mod + 1) for = 2).

Result 5.2.1 ([12, Corollary 3.9, Theorem 5.1]). Let t ∈ N \ {0} and r = 3t+ 1. Suppose thatq = (q′)3, with q > 64 if t > 1. Then

ℓq(r, 3) 6 4(q′)r−3 + 4(q′)r−4, and µq(3) <32

3+

32

q′+

32

(q′)2− 64

3q.

Result 5.2.2 ([12, Theorems 3.16 and 5.2]). Let t ∈ N \ {0} and r = 3t + 2. Suppose thatq = (q′)3, with q > 27 if t > 1. Then

ℓq(r, 3) 6 9(q′)r−3 − 8(q′)r−4 + 4(q′)r−5, and µq(3) <243

2− 324

q′+

72

(q′)2.

Our results (Corollary 7.2.11) imply that

ℓq(r, 3) 6 6(q′)r−2 − 1

q′ − 1, and µq(3) < 36

(1 +

1

q′+

1

(q′)2

)3

(3)

and hence do not improve Result 5.2.1 if r = 4 or if r = 3t + 1, t > 1 and q′ > 4. However, incase r = 3t+2, the upper bound on ℓq(r, 3) in (3) does improve Result 5.2.2 if q′ > 5; the upperbound on µq(3) improves Result 5.2.2 if q′ > 7.

More generally, the authors of [12] presented the following.

8

Result 5.2.3 ([12, Theorem 3.15]). Let N > +1 and suppose q = (q′)+1 for any prime powerq′. Then

sq(N, ) 6

∑N−+1i=0 (q′ − 1)i

(N+1i

)− 1

q′ − 1∼(N + 1

)(q′)N−.

At first sight, our results (Theorem 7.2.10) present a significant improvement on the bound

presented in Result 5.2.3, as the binomial coefficient(N+1

)is reduced to (+1)(+2)

2 . In fact,Davydov et al. [12] speculated that their coefficient could be improved, as they mention thisas an open problem. However, a certain degree of nuance is needed, as the following resultspresent better bounds in case r 6≡ 0 (mod R), R > 4, and r is large enough (equivalently, incase N + 1 6≡ 0 (mod + 1), > 3, and N is large enough).

Result 5.2.4 ([12, Theorem 6.3]). Let t ∈ N and r = Rt+ 1 (equivalently, N = (+ 1)t) withR > 4 (equivalently, > 3). Suppose that q = (q′)R = (q′)+1, q′ > 4, and choose t0 ∈ N suchthat

qt0−1 > (q′ − 1)

(R(R+ 1)

2− 2

)+R+ 5.

If t > t0, then there exists an [n, n− r]qR code with

n =

(R(R+ 1)

2− 2

)(q′)r−R −

((R − 1)R

2− 7

)(q′)r−R−1, hence ℓq(r,R) 6 n.

This code corresponds to a -saturating set of PG(N, q), with N = r − 1 and = R− 1, of size

n =

((+ 1)(+ 2)

2− 2

)(q′)N− −

((+ 1)

2− 7

)(q′)N−−1, hence sq(N, ) 6 n.

Whether we compare this result with Theorem 7.2.10 or Corollary 7.2.11, one can easily see thatour results do not improve Result 5.2.4 for given restrictions on r, R (equivalently, on N , ), tand q. Notice that we omitted the case t = 1 from the original result. However, we discuss thisparticular case in Remark 5.2.10.

Result 5.2.5 ([12, Theorem 6.4]). Let t ∈ N and r = Rt+γ (equivalently, N+1 = (+1)t+γ),γ ∈ {2, 3, . . . , R − 1} with R > 4 (equivalently, > 3). Suppose that q = (q′)R = (q′)+1 andchoose t0 ∈ N such that

qt0−1 > n(γ)R,q, with n

(γ)R,q :=

γ∑

i=0

(q′ − 1)i(R+ γ

i+ 1

)∼(R+ γ

R− 1

)(q′)γ .

If t = 1 or t > t0, then there exists an [n, n− r]qR code with

n = n(γ)R,q · (q′)r−R−γ + w

(q′)r−R−γ − 1

q − 1, 0 6 w 6 R− 3, hence ℓq(r,R) 6 n.

This code corresponds to a -saturating set of PG(N, q), with N = r − 1 and = R− 1, of size

n = n(γ)+1,q · (q′)N−−γ + w

(q′)N−−γ − 1

q − 1, 0 6 w 6 − 2, hence sq(N, ) 6 n.

In general, our results (Theorem 7.2.10 or Corollary 7.2.11) improves the result above; thisimprovement increases as γ grows. Moreover, the authors of [12] note that the main term of theasymptotic covering density of the infinite family of covering codes arising from Result 5.2.5 is

equal to((R+γ)R−1

(R−1)!

)R· 1R! , which is significant larger than

(R(R−1)

)RR! (see Corollary 7.2.11).

9

Remark 5.2.6. Davydov, Giulietti, Marcugini and Pambianco [12, Section 7] cleverly extendedthe results above. More specifically, if R′ is a proper divisor of R, r = Rt+s R

R′ (s ∈ {1, 2, . . . , R′−1}) and q = (q′)R

′

, they managed to construct infinite families covering codes of length roughly

equal to RR′

(R′+sR′−1

)q

r−RR if t = 1 or t is large enough (Note that Result 5.1.3 exists under the same

conditions, for R′ = 2). Under these conditions, our results present improvement if q is an Rth

power, R′ > 4 and

1. either t > 1 is relatively small, or

2. R+12 < 1

R′

(R′+sR′−1

).

The latter condition is true if R′ is a relatively large divisor of R (R′ & s s√R). In all other cases,

the results of [12, Section 7] are better than ours.We would shortly want to point out a misprint in [12, Corollary 7.5]. It should state thatγ ∈ {2, 3, . . . , R′ − 1} instead of γ ∈ {2, 3, . . . , R− 1}.

Lastly, Result 5.2.1 arose by cleverly choosing four disjoint lines in PG(3, q′) of which the unionof points forms a 3-fold strong blocking set (and by subsequently qm-concatenating the coveringcode arising from the obtained 2-saturating set). This idea of choosing pairwise disjoint (N−)-spaces of PG(N, q′), of which the union of points forms a (+1)-fold strong blocking set, led tothe following, more general results if q′ is large enough.

Result 5.2.7 ([19, Theorem 24] and [20, Proposition 10, Subsection 3.4]).

1. Let q = (q′)N , q′ > 2N − 1. Then

sq(N,N − 1) 6 (2N − 1)(q′ + 1).

2. Let 1 < < N and q = (q′)+1, q′ > N + 1. Then

sq(N, ) 6((N − + 1)+ 1

)(q′)N−+1 − 1

q′ − 1.

If q′ > N + 1, one can check that our results (Theorem 7.2.10) improve Result 5.2.7(2.) if andonly if < 2N−1

3 . We discuss the comparison between our results and Result 5.2.7(1.) in Remark5.2.10.

The mixed subgeometry approach

The mixed subgeometry approach is based on constructing saturating sets as a union of severaldistinct subgeometries which are not part of a common, larger subgeometry. This approachwas the main source of inspiration for this article. The technique is used much less than thestrong blocking set approach. In fact, Result 5.2.8 below is the only instance using the mixedsubgeometry approach that we encountered in the literature.

Result 5.2.8 ([10, Theorem 5.2]). Let q be square. Let b1, b2 and b3 be three distinct Baersublines spanning PG(2, q) and sharing a common point P , with the addition that b1 and b2share a further point Q 6= P as well. Then (b1 ∪ b2 ∪ b3) \ {P} is a 1-saturating set of PG(2, q).As a consequence,

sq(2, 1) 6 3√q − 1.

Better bounds on sq(2, 1), q square, are known (see [14, Proposition 9] for an overview, whichincludes improvements arising from Result 5.1.3). Interestingly, as noted in [14, Remarks 3 and4], if q equals the square of a prime number, no better bound on sq(2, 1) than the one depictedin Result 5.2.8 is known in the literature. Our interest was mainly peaked by the underlying

10

(sub)geometric construction. In fact, Construction 7.2.4 is basically a highly generalised versionof the construction described in Result 5.2.8.

By making use of variations of qm-concatenating constructions, the following bound is obtained,generalising the bound of Result 5.2.8.

Result 5.2.9 ([11, Example 6, Equation (33)]). Let N be even and q > 16 be square. Then

sq(N, 1) 6 (3√q − 1)q

N2−1 +

⌊q

N2−2⌋.

Theorem 7.2.9 implies, for N even and q square, that

sq(N, 1) 6 3√q ·(q

N2−1 + q

N2−2 + · · ·+ 1

)− N

2.

Hence, in case = 1, we only achieve improvement if q ∈ {4, 9}.

Remark 5.2.10 (The case = N − 1). Observe the following:

(A) One can illustrate the use of the strong blocking set approach by considering three non-concurrent lines of a Baer subplane, proving that sq(2, 1) 6 3

√q if q is square.

(B) On the other hand, the mixed subgeometry approach led to Result 5.2.8, which states thatsq(2, 1) 6 3

√q − 1 if q is square, hence improving the bound of (A) by 1.

Curiously, if q = (q′)N , we discover the same phenomenon when observing a specific general-isation of the construction behind each of these two bounds. On the one hand, Davydov andOstergard [16, Theorem 6] generalised (A) by constructing the so-called tetrahedron, obtainedby connecting N + 1 points of PG(N, q′) in general position. This gives rise to the expression

sq(N,N − 1) 6N(N + 1)

2q′ − N(N − 1)

2+ 1. (4)

On the other hand, the construction behind Theorem 7.2.9 (mixed subgeometry approach) gen-eralises (B) and, if N > 1, gives rise to

sq(N,N − 1) 6N(N + 1)

2q′ − N(N − 1)

2, (5)

which improves (4) yet again by 1.Although this phenomenon is somewhat curious, there are better upper bounds known for =N − 1, q = (q′)N , N > 3. If q′ > N + 1, Result 5.2.7(2.) generally improves (5); moreover, [12,Corollary 3.12] states that

sq(N,N − 1) 6N(N + 1)

2q′ − N(N − 1)

2− 2q′ + 7,

which clearly improves (5) if q′ > 4. In conclusion, our results only improve the case = N − 1if N > 3 and q′ ∈ {2, 3}.

6 The geometries Y (,m, q′) and T ∗(D,m,q′)

In this section, we put the topic of saturating sets temporarily on hold. We will focus onan isomorphism between a point-line geometry Y (,m, q′) (see Definition 6.1.1) and the linearrepresentation T ∗(D,m,q′) (see Definition 6.1.2) of a -dimensional q′-subgeometry D,m,q′ ofPG(+ 1, (q′)m

)(m ∈ N \ {0}). This will be done by taking advantage of a coordinate system

11

of the respective ambient projective spaces (although an alternative exists using field reduction,see Subsection 6.4).

We have reason to believe that the point-line geometry Y (,m, q′) hasn’t been considered in theliterature before.2

6.1 Preliminaries

We introduce two point-line geometries. Pay attention to the fact that within the first point-linegeometry, points of PG

(m, (q′)+1

)are considered, while in the second point-line geometry we

consider points of PG(+ 1, (q′)m

).

Definition 6.1.1. Let m ∈ N \ {0}. Consider an (m − 1)-dimensional q′-subgeometry C ofPG(m, (q′)+1

)and define ΣC := 〈C〉(q′)+1 . The point-line geometry Y (,m, q′) is the incidence

structure (PC ,LC) with natural incidence, where

• PC is the set of points of PG(m, (q′)+1

)\ ΣC, and

• LC is the set of all point sets B \ C, where B is an m-dimensional q′-subgeometry ofPG(m, (q′)+1

)that contains C.

If PG(m, (q′)+1

)is embedded in PG

(m′, (q′)+1

)(m′ > m) as an m-dimensional subspace Π,

we will use notation PΠC and LΠ

C , respectively, to avoid confusion when considering more thanone such point-line geometry.

Secondly, we brush up the concept of linear representations. This notion was independentlyintroduced for hyperovals by Ahrens and Szekeres [1] and Hall [21], and extended to generalpoint sets by De Clerck [17].

Definition 6.1.2. Let m ∈ N \ {0}. Consider a point set K of PG(, (q′)m

)embedded in

PG(+1, (q′)m

). The linear representation of K is the point-line geometry T ∗(K) := (PK,LK)

with natural incidence, where

• PK is the set of points of PG(+ 1, (q′)m

)\ PG

(, (q′)m

), and

• LK is the set of all point sets ℓ \ {P}, where ℓ * PG(, (q′)m

)is a line of PG

(+1, (q′)m

)

intersecting K in a point P .

6.2 A direct isomorphism between Y (,m, q′) and T ∗(D,m,q′) using coordi-nates

Consider the following configuration.

Configuration 6.2.1. Let m ∈ N \ {0} and suppose q = (q′)+1. Consider an (m − 1)-dimensional q′-subgeometry C of PG(m, q) and define ΣC := 〈C〉q. Choose a coordinate systemfor PG(m, q) such that

• E1, . . . , Em are the points with coordinates (0, 1, 0, . . . , 0), . . . , (0, 0, 0, . . . , 1), respectively,

• E′ is the point with coordinates (0, 1, . . . , 1), and

• C is the (by Lemma 2.4 unique) (m−1)-dimensional q′-subgeometry containing the pointsE1, . . . , Em and E′.

2With exception of the case m = 1, as there exists a well-known isomorphism between the affine parts ofq′-sublines of PG

(

1, (q′)+1)

through a fixed point (which can be viewed as q′-sublines of AG(

1, (q′)+1)

) and thelines of AG(+ 1, q′).

12

Furthermore, let D,m,q′ be a -dimensional q′-subgeometry of PG( + 1, (q′)m

)and define

ΣD,m,q′:=⟨D,m,q′

⟩(q′)m

. Choose a coordinate system for PG(+ 1, (q′)m

)such that

• D,m,q′ is the (by Lemma 2.4 unique) -dimensional q′-subgeometry containing all pointscorresponding to the set of coordinates {(0, 1, 0, . . . , 0), . . . , (0, 0, 0, . . . , 1), (0, 1, 1, . . . , 1)}.

For notational simplicity, we will often write D instead of D,m,q′ .

Lemma 6.2.2. Consider Configuration 6.2.1. Let P,Q /∈ ΣC be two distinct points of PG(m, q)with coordinates (1, x1, x2, . . . , xm) and (1, y1, y2, . . . , ym) (xi, yi ∈ Fq), such that PQ intersectsΣC in E′. Let B be the (by Lemma 2.5 unique) m-dimensional q′-subgeometry containing C, Pand Q. Then the set of coordinates of all points in B \ C is equal to

{(1, x1 + k1(y1 − x1), x2 + k2(y1 − x1), . . . , xm + km(y1 − x1)

): k1, . . . , km ∈ Fq′

}.

Proof. It is clear that the hyperplane ΣC is defined by the equation X0 = 0. Suppose thatE0 is the point of PG(m, q) with coordinates (1, 0, 0, . . . , 0) and E the point with coordinates(1, 1, 1, . . . , 1), and let B0 be the (by Lemma 2.4 unique) m-dimensional q′-subgeometry con-taining the frame {E0, . . . , Em, E}. As this is the canonical frame, it is clear that the set ofcoordinates of all points in B0 \ ΣC is equal to

{(1, k1, k2, . . . , km) : k1, . . . , km ∈ Fq′

}.

One can find an element of PGL(m + 1, q) that maps the canonical frame {E0, E1, . . . , Em, E}onto the frame {P,E1, . . . , Em, Q}, which can be represented by an Fq-multiple of the followingmatrix:

1 0 0 · · · 0x1 y1 − x1 0 · · · 0x2 0 y2 − x2 · · · 0...

......

. . ....

xm 0 0 · · · ym − xm

.

Such a matrix maps a point of B0 with coordinates (1, k1, k2, . . . , km), ki ∈ Fq′ , onto a point ofB with coordinates

(1, x1 + k1(y1 − x1), x2 + k2(y2 − x2), . . . , xm + km(ym − xm)

).

Note that, as E′ ∈ PQ, the tuple (0, y1 − x1, y2 − x2, . . . , ym − xm) has to be an Fq-multiple of(0, 1, 1, . . . , 1), which implies that yi − xi = yj − xj for all i, j ∈ {1, 2, . . . ,m}. Hence, the set ofcoordinates of all points in B \ C can be simplified to

{(1, x1 + k1(y1 − x1), x2 + k2(y1 − x1), . . . , xm + km(y1 − x1)

): k1, . . . , km ∈ Fq′

}.

We now introduce the following map ϕ.

Definition 6.2.3. Consider Configuration 6.2.1. Choose elements α ∈ Fq and β ∈ F(q′)m suchthat Fq′ [α] ∼= Fq and Fq′ [β] ∼= F(q′)m . Define the map ϕ : PC → PD that maps a point of PCwith coordinates

(1, z1, z2, . . . , zm) =

1,

∑

j=0

z1jαj ,

∑

j=0

z2jαj , . . . ,

∑

j=0

zmjαj

(zk ∈ Fq, zrs ∈ Fq′)

onto the unique point of PD with coordinates(1,

m∑

i=1

zi0βi−1,

m∑

i=1

zi1βi−1, . . . ,

m∑

i=1

ziβi−1

).

If PG(m, q) is embedded in a larger projective geometry as a subspace Π, we will use the notationϕ|Π to clarify which map is considered.

13

Theorem 6.2.4. Let m ∈ N \ {0}. Then the point-line geometries Y (,m, q′) and T ∗(D,m,q′)are isomorphic.

Proof. Let q = (q′)+1. Note that the choice of coordinates made in Configuration 6.2.1 does notaffect the generality of the theorem. After all, any collineation of PG(m, q) preserves elementsof LC as being (the affine parts of) m-dimensional q′-subgeometries containing the image of C,hence the whole set LC is preserved and, furthermore, incidence is sustained. The same holdsfor the point-line geometry T ∗(D,m,q′).Thus, consider Configuration 6.2.1. Let B \ C be an arbitrary element of LC . Suppose thatP,Q ∈ B \ C are two distinct points with coordinates (1, x1, x2, . . . , xm) and (1, y1, y2, . . . , ym)(xi, yi ∈ Fq), such that PQ intersects ΣC in E′. By Lemma 2.5, B is uniquely defined by C, Pand Q. By Lemma 6.2.2, the set of coordinates of all points in B \ C is equal to

{(1, x1 + k1(y1 − x1), x2 + k2(y1 − x1), . . . , xm + km(y1 − x1)

): k1, . . . , km ∈ Fq′

}.

Consider the map ϕ (see Definition 6.2.3); note that ϕ is a bijection, as one can easily defineits inverse. We will prove that ϕ induces an isomorphism between the point-line geometriesY (,m, q′) and T ∗(D,m,q′).If xi =

∑j=0 xijα

j and yi =∑

j=0 yijαj (xij , yij ∈ Fq′), then the set of coordinates of the images

of all points in B \ C under ϕ is equal to{(

1,

m∑

i=1

(xi0 + ki(y10 − x10)

)βi−1, . . . ,

m∑

i=1

(xi + ki(y1 − x1)

)βi−1

): k1, . . . , km ∈ Fq′

}

=

{(1,

m∑

i=1

xi0βi−1, . . . ,

m∑

i=1

xiβi−1

)

+

m∑

i=1

kiβi−1 · (0, y10 − x10, . . . , y1 − x1) : k1, . . . , km ∈ Fq′

}

=

{(1,

m∑

i=1

xi0βi−1, . . . ,

m∑

i=1

xiβi−1

)+ k · (0, y10 − x10, . . . , y1 − x1) : k ∈ F(q′)m

}. (6)

The latter set is equal to the set of coordinates of all points on l \ ΣD, with l a line of PG(+

1, (q′)m)through ϕ(P ) /∈ ΣD intersecting ΣD in the point of D with coordinates (0, x10 −

y10, . . . , x1 − y1) ⊆ F+2q′ . Hence, as it is clear that ϕ maps points on a line of Y (,m, q′) onto

points on a line of T ∗(D), this map naturally induces a morphism from Y (,m, q′) to T ∗(D).As ϕ is a bijection between PC and PD, this map is injective w.r.t. the line sets LC and LD,hence the only thing left to prove is the fact that ϕ is surjective w.r.t. these line sets.It is clear that an element l \ {D′} of LD is uniquely defined by the point D′ ∈ D and apoint P ′ ∈ l \ {D′}. By observing (6), it is clear that we can fix the point P := ϕ−1(P ′) ∈PG(m, q) \ ΣC which has, let us say, coordinates (1, x1, . . . , xm) ∈ Fm+1

q , and try to choose a

point Q ∈ PG(m, q) \ ΣC such that

1. PQ intersects ΣC in a point of C, and

2. the element of LC defined by C, P and Q is mapped by ϕ onto a line intersecting ΣD inD′ ∈ D.

Condition 1 is clearly fulfilled if we choose a point Q ∈ PG(m, q) with coordinates (1, y1, . . . , ym)such that y1 − x1 = y2 − x2 = · · · = ym − xm. These equations mean that, once we fix thevalue y1 ∈ Fq, we fix the entire point Q. This implies that we have freedom to choose anyvalue y1 =

∑j=0 y1jα

j ∈ Fq to try and satisfy condition 2. By observing (6), it is clear that this

freedom of choice implies that we can reach each tuple of coordinates in {0}×F+1q′ corresponding

to a point of D, in particular the coordinates of D′ ∈ D.

14

6.3 Notions of parallelism and independence in Y (,m, q′)

Theorem 6.2.4 states that the point-line geometry Y (,m, q′) is isomorphic to the linear repre-sentation T ∗(D,m,q′) of a -dimensional q′-subgeometry D,m,q′ of PG

(, (q′)m

)(see Definition

6.1.1 and Definition 6.1.2). As the lines of a linear representation are embedded in an affinegeometry, notions of parallelism and independence of concurrent lines seem transferable to thepoint-line geometry Y (,m, q′).

Definition 6.3.1. Let m ∈ N \ {0} and consider the point-line geometry Y (,m, q′). Then

• distinct lines of LC are called concurrent if they contain a common point.

Now define ϕ := ϕ ◦ ϕY , where ϕY is a collineation of PG(m, q) such that the image of thepoint-line geometry Y (,m, q′) under ϕY corresponds to the coordinate system described inConfiguration 6.2.1, and ϕ is the isomorphism described in Definition 6.2.3. Then

• two lines of LC are said to be ϕ-parallel if their images under ϕ are parallel,

• concurrent lines of LC are said to be ϕ-independent if their images under ϕ are indepen-dent3,

• we will call (the point set of) any line of LC a 1-dimensional affine ϕ-subspace. Recursively,for any d ∈ {2, . . . , + 1}, a union of (q′)(d−1)m ϕ-parallel lines of LC is said to be a d-dimensional affine ϕ-subspace if each of these lines intersects a fixed (d − 1)-dimensionalaffine ϕ-subspace in precisely one point.

Lemma 6.3.2. The notions described in Definition 6.3.1 are well-defined.

Proof. Assume that Y (,m, q′) is chosen in such a way that it corresponds to the coordinatesystem described in Configuration 6.2.1. For this lemma to be true, we want to prove that ifone of the last three notions described in Definition 6.3.1 holds for certain points or lines ofY (,m, q′), it still holds for the images of these points or lines w.r.t. any collineation of theambient geometry PG(m, q).Let B \C be an arbitrary element of LC and P,Q ∈ B\C be two distinct points with coordinates(1, x1, x2, . . . , xm) and (1, y1, y2, . . . , ym), respectively (xi, yi ∈ Fq), such that PQ intersects ΣCin E′ (see Configuration 6.2.1), implying that

y1 − x1 = · · · = ym − xm. (7)

Assuming that xi =∑

j=0 xijαj and yi =

∑j=0 yijα

j (Fq′ [α] ∼= Fq and xij , yij ∈ Fq′ , seeDefinition 6.2.3), just as in (6), the set of coordinates of the images of all points in B \ C underϕ is equal to

{(coordinates of ϕ(P )) + k · (0, y10 − x10, . . . , y1 − x1) : k ∈ F(q′)m

}.

For any element B \ C of LC , we can analogously choose two distinct points P ,Q ∈ B \ C withcoordinates (1, x1, x2, . . . , xm) and (1, y1, y2, . . . , ym), respectively (xi, yi ∈ Fq), such that PQintersects ΣC in E′, implying that

y1 − x1 = · · · = ym − xm. (8)

The set of coordinates of the images of all points in B \ C under ϕ is equal to

{(coordinates of ϕ(P )

)+ k · (0, y10 − x10, . . . , y1 − x1) : k ∈ F(q′)m

}.

3Concurrent lines ℓ1, ℓ2, . . . , ℓd of a projective space are called independent if they span a subspace of dimensiond.

15

Hence, B \ C and B \ C are ϕ-parallel if and only if there exists an element γ ∈ F(q′)m suchthat (0, y10 − x10, . . . , y1 − x1) = γ · (0, y10 − x10, . . . , y1 − x1). Moreover, as both of these

coordinates are tuples of F+2q′ , γ has to be an element of Fq′ . Combining this with (7) and (8),

we get that yij − xij = γ(yij − xij) for a γ ∈ Fq′ independent of i or j, implying that

(y1 − x1, . . . , ym − xm) = γ · (y1 − x1, . . . , ym − xm) for a γ ∈ Fq′ . (9)

In conclusion, two elements B \ C and B \ C are ϕ-parallel if and only if there exist two points Pand Q in B \ C on a line through E′ and two points P and Q in B \ C on a line through E′ suchthat property (9) holds for their respective coordinates. As Fq′ is fixed under any automorphismof the ambient field Fq, we can observe that property (9) stays valid if P , Q, P and Q are movedby any collineation of PG(m, q). This implies that ϕ-parallelism is invariant w.r.t. collineations,hence this notion is well-defined.Using this, we can prove the same for the notion of a d-dimensional affine ϕ-subspace (d ∈{1, . . . , + 1}). If d = 1, this is trivially true. Note that any such a d-dimensional affineϕ-subspace is a set of (q′)dm points of PC . If d > 2, such a d-dimensional affine ϕ-subspaceoccurs as a union of ϕ-parallel lines through each of the points of a (d − 1)-dimensional affineϕ-subspace. As any collineation of PG(m, q) preserves incidence, ϕ-parallelism and, inductively,(d− 1)-dimensional affine ϕ-subspaces, the proof follows.Finally, as the points of d concurrent, ϕ-independent lines of LC are contained in a uniqued-dimensional affine ϕ-subspace, the invariance of the latter affine ϕ-subspace implies the in-variance of the ϕ-independence of those lines.

Consider the following configuration.

Configuration 6.3.3. Let m ∈ N \ {0} and suppose q = (q′)+1. Consider an (m − 2)-dimensional q′-subgeometry C of PG(m, q) and define ΣC := 〈C〉q. Let Π1, Π2 and Π3 be threedistinct hyperplanes of PG(m, q) through ΣC .

Lemma 6.3.4. Consider Configuration 6.3.3. Let B ⊇ C be an (m − 1)-dimensional q′-subgeometry of PG(m, q) such that 〈B〉q = Π1 and consider a point S ∈ Π2 \ ΣC. Then thereexists a unique m-dimensional q′-subgeometry A containing B and S and intersecting Π3 \ ΣC.

Proof. Take a point R ∈ B \ C. Then the line RS has to intersect Π3 in a point T /∈ ΣC . Anym-dimensional q′-subgeometry that contains B and S and intersects Π3 in a point outside of ΣC ,has to contain T and, in particular, the (by Lemma 2.4) unique q′-subline defined by R, S andT . By Lemma 2.5, there exists exactly one such q′-subgeometry.

Definition 6.3.5. Consider Configuration 6.3.3. Then we can define, for any point S ∈ PΠ2C ,

the projection mapprojSΠ1,Π3

: LΠ1C → LΠ3

C : B \ C 7→ (A ∩Π3) \ C,and the shadow map

shadSΠ1,Π3: LΠ1

C → LΠ2C : B \ C 7→ (A ∩Π2) \ C,

with A the (by Lemma 6.3.4) unique m-dimensional q′-subgeometry containing B and S andintersecting Π3 \ΣC . Furthermore, for a fixed element B \ C ∈ LΠ1

C , we can naturally extend the

definition above and define, for any subset T ⊆ PΠ2C ,

projTΠ1,Π3(B \ C) :=

⋃

S∈TprojSΠ1,Π3

(B \ C)

andshadTΠ1,Π3

(B \ C) :=⋃

S∈TshadSΠ1,Π3

(B \ C).

16

Lemma 6.3.6. Consider Configuration 6.3.3. Let B \ C ∈ LΠ1C and S1, S2 ∈ PΠ2

C . Then

shadS1Π1,Π3

(B \ C) and shadS2Π1,Π3

(B \ C) are ϕ|Π2-parallel.

Proof. Choose coordinates for PG(m, q) and consider the points

E0(1, 0, 0, . . . , 0), E1(0, 1, 0, . . . , 0), . . . , Em(0, 0, . . . , 0, 1)

andE(1, 1, 1, . . . , 1), E′(0, 1, 1, . . . , 1) and E′′(0, 0, 1, . . . , 1).

By Lemma 6.3.2 we can assume, without loss of generality, that

• C is (by Lemma 2.4) uniquely defined by the points E2, . . . , Em and E′′,

• E1 is a point of B, and

• E0 and E are the points l ∩Π2 and l ∩Π3, respectively, with l * Π1 an arbitrarily chosenline intersecting Π1 in a point of B \ (C ∪ {E1}).

In this way, B is (indirectly by Lemma 2.5) uniquely defined by C, E1 and E′ ∈ E0E = l.Assuming S1 has coordinates (1, 0, x2, . . . , xm) (xi ∈ Fq), the line E′S1 intersects Π3 in a pointT1 with coordinates (1, 1, 1 + x2, . . . , 1 + xm).By Lemma 6.3.4, there exists a unique m-dimensional q′-subgeometry A containing B, S1 andT1. By Lemma 6.2.2, the set of coordinates of all points in A \ B is equal to

{(1, k1, x2 + k2, . . . , xm + km

): k1, . . . , km ∈ Fq′

}.

As a consequence, the set of coordinates of all points of shadS1Π1,Π3

(B \ C) is equal to{(

1, 0, x2 + k2, . . . , xm + km): k2, . . . , km ∈ Fq′

}. (10)

Restricting these coordinates to the geometry PG(m− 1, q) ∼= Π2 (hence by ignoring the secondcoordinate 0), the set of coordinates of the images of all points (10) under ϕ|Π2

is, as in (6),equal to {(

coordinates of ϕ|Π2(S1)

)+ k · (0, 1, 0, . . . , 0) : k ∈ F(q′)m−1

}.

As the line parallel class of the affine line that arises in this way does not rely on the choice ofthe point S1 ∈ PΠ2

C , the lemma is proven.

Lemma 6.3.7. Consider Configuration 6.3.3. Let B1 \ C,B2 \ C, . . . ,Bj \ C be j distinct, ϕ|Π1-

independent elements of LΠ1C sharing a point F ∈ PΠ1

C (j ∈ {1, 2, . . . , }) and suppose T ⊆ PΠ2C

is a (j − 1)-dimensional affine ϕ|Π2-subspace. Then there exists a k ∈ {1, 2, . . . , j} such that

projTΠ1,Π3(Bk \ C) is a j-dimensional affine ϕ|Π2

-subspace.

Proof. Choose a point F ′ ∈ T and define F ′′ := FF ′ ∩ Π3. The projection of points of Π1

onto Π2 through the point F ′′ is a natural projectivity between the spaces when interpreted asdistinct projective geometries. Hence, if one projects B1 \ C,B2 \ C, . . . ,Bj \ C onto Π2 in thisway, we obtain j distinct, ϕ|Π2

-independent elements B′1 \ C,B′

2 \ C, . . . ,B′j \ C of LΠ2

C sharingthe point F ′ ∈ T . As T is a (j − 1)-dimensional affine ϕ|Π2

-subspace, there has to exist a

B′k \ C ∈ LΠ2

C which has only the point F ′ in common with T . Moreover, it is easy to see

that B′k \ C = shadF

′

Π1,Π3(Bk \ C). Hence, by Lemma 6.3.6, shadTΠ1,Π3

(Bk \ C) is a union of |T |distinct, ϕ|Π2

-parallel elements of LΠ2C , each containing a unique point of T . In other words,

shadTΠ1,Π3(Bk \ C) is a j-dimensional affine ϕ|Π2

-subspace. By considering the natural projection

of points of Π2 onto Π3 through F , one can easily check that shadTΠ1,Π3(Bk \ C) gets projected

onto projTΠ1,Π3(Bk \ C).

17

6.4 A detour Y (,m, q′) ∼ X(,m, q′) ∼ T ∗(D,m,q′) using field reduction

In Subsection 6.2, we successfully obtained an isomorphism between Y (,m, q′) and T ∗(D,m,q′)using coordinates. The notation Y (,m, q′), however, was chosen for a reason, as we want toemphasize its link with the point-line geometry X(,m, q′) introduced by De Winter, Rotteyand Van de Voorde.

Definition 6.4.1 ([18, Section 3]). Let m ∈ N \ {0}. Consider a -dimensional subspace π ofPG( + m, q′). The point-line geometry X(,m, q′) is the incidence structure (PX ,LX) withnatural incidence, where

• PX is the set of all (m− 1)-spaces of PG(+m, q′) disjoint to π, and

• LX is the set of all m-spaces of PG(+m, q′) meeting π exactly in one point.

In their work, the authors construct an explicit isomorphism betweenX(,m, q′) and T ∗(D,m,q′).

Theorem 6.4.2 ([18, Theorem 4.1]). Let m ∈ N \ {0}. Let D,m,q′ be a -dimensional q′-subgeometry of PG

(, (q′)m

). Then the point-line geometries X(,m, q′) and T ∗(D,m,q′) are

isomorphic.

This theorem, together with Theorem 6.2.4, implies that Y (,m, q′) ∼ X(,m, q′).

As a side note, we would like to mention that it is possible to work the other way around byconstructing a direct isomorphism between Y (,m, q′) and X(,m, q′) (different from the com-position of the isomorphisms behind Theorem 6.2.4 and Theorem 6.4.2) by using field reduction(see [23] for an introduction to this technique). Although this results in a more elegant isomor-phism between these two point-line geometries, we didn’t include it in this work as we needed anexplicit isomorphism describing the link between Y (,m, q′) and T ∗(D,m,q′) to exploit notionsof parallelism and independence of concurrent lines (see Subsection 6.3). Besides, the descriptionand proof of this isomorphism easily takes up several pages.

7 Constructing saturating sets using subgeometries

In this section, we switch our focus back to saturating sets. We will construct a point set (seeConstruction 7.2.4) in PG(N, q), q = (q′)+1, and prove that this is a -saturating set. Theexistence of this saturating set directly leads to the main result of this article, namely thatsq(N, ) . (+1)(+2)

2 (q′)N− (see Theorems 7.2.9 and 7.2.10).

7.1 Preliminaries

We will make use of the following concept.

Definition 7.1.1. Let m ∈ N \ {0}. Let s ∈ N, let t ∈ N \ {0, 1} and consider an (s − 1)-subspace Σ in PG(m, q). A set of independent4 s-subspaces F := {τ1, . . . , τt} through Σ iscalled an s-flower with pistil Σ. The elements of F are called the petals of the s-flower.

Furthermore, we will at one point need the following property concerning q′-subgeometries.

Lemma 7.1.2. Let m ∈ N \ {0} and suppose q = (q′)+1. Consider an (m − 1)-dimensionalq′-subgeometry C1 of PG(m, q) and an (m−2) q′-subgeometry C2 ⊆ C1; define Σi := 〈Ci〉q. Let B1

and B2 be two distinct m-dimensional q′-subgeometries, both containing C1 and a point F /∈ Σ1,and suppose that Π is an (m− 1)-space through Σ2 not equal to Σ1 and not containing F . ThenΠ cannot intersect both B1 and B2 in an (m− 1)-dimensional q′-subgeometry.

4Alternatively, one can state that dim (τ1 ∩ · · · ∩ τt) = s− 1 and dim (〈τ1, . . . , τt〉) = s+ t− 1.

18

Proof. Suppose that the contrary is true. Choose a point F ′ ∈ C1 \ C2. Then the line FF ′

intersects Π in a point P . As Π intersects both B1 and B2 in a q′-subgeometry of maximaldimension, P has to be a point of both B1 as B2. Moreover, as both these subgeometries containC1 ∋ F ′ and F , the unique q′-subline containing F , F ′ and P has to be contained in both B1

and B2. By Lemma 2.5, this would imply that B1 = B2, a contradiction.

Finally, the following lemma is a direct consequence of the strong blocking set approach (seeSubsection 5.2).

Lemma 7.1.3 ([12, Section 3]). Let m ∈ N and let q = (q′)+1. Then any m-dimensionalq′-subgeometry of PG(m, q) is a -saturating set of PG(m, q).

Proof. Let B be an m-dimensional q′-subgeometry of PG(m, q). If m 6 , we can simply choosea base of the subgeometry B; naturally, such a set of points spans the whole space PG(m, q).As this base contains m+ 1 6 + 1 points, the proof is done.If m > + 1, then the proof is exactly the same as described in [12, proof of Theorem 3.2].

7.2 Constructing the saturating set

We will construct a small -saturating set of PG(N, q) by making use of the following observation.

Lemma 7.2.1. Let m ∈ N \ {0}, 0 < < m and q = (q′)+1. Suppose that F := {τ1, . . . , τ+1}is an (m− )-flower of PG(m, q) with certain pistil Σ. Let C ⊆ Σ be an (m− − 1)-dimensionalq′-subgeometry and consider, for every j ∈ {1, 2, . . . , +1}, j distinct, ϕ|τj -independent elements

B(1)j \ C,B(2)

j \ C, . . . ,B(j)j \ C of Lτj

C , all sharing a point Fj ∈ PτjC . Then the point set

B :=

+1⋃

j=1

j⋃

k=1

(B(k)j \ C

)

-saturates all points of PG(m, q) that do not lie in the span of any petals of F .

Proof. Let P be a point not contained in the span of any petals of F . Define, for everyj ∈ {1, 2, . . . , + 1}, Πj := 〈τj , τj+1, . . . , τ+1〉. Note that, by definition of a flower (Definition7.1.1), Π1 is equal to the whole space PG(m, q), hence P ∈ Π1. Furthermore, consider the(m − )-space π0 := 〈Σ, P 〉 and the point set T0 := {P}. One can now iterate through thefollowing process, for j going from 1 to .

1. Observe that πj−1 and τj are distinct (m− )-subspaces through Σ, contained in Πj butnot contained in Πj+1. As the latter is a hyperplane of Πj , 〈πj−1, τj〉 intersects Πj+1 inan (m− )-space πj := 〈πj−1, τj〉 ∩Πj+1.

2. Observe that πj−1, τj and πj are three distinct (m − )-spaces through Σ, spanning an(m − + 1)-space. Furthermore, Tj−1 ⊆ Pπj−1

C is a (j − 1)-dimensional affine ϕ|πj−1-

subspace. By Lemma 6.3.7, there exists a B(k)j \C in Lτj

C such that Tj := projTj−1τj ,πj(B(k)

j \C)is a j-dimensional affine ϕ|πj

-subspace.

3. Note that any point Tj ∈ Tj lies in the span of a point of B ∩ τj and a point of Tj−1.

Indeed, by definition of projTj−1τj ,πj , there has to exist a point T ′

j ∈ Tj−1 such that Tj ∈proj

T ′

jτj ,πj(B(k)

j \ C). Hence, there exists a point of B(k)j \ C that is projected from T ′

j ontoTj.

Eventually, T ⊆ Pπ

C is a -dimensional affine ϕ|π-subspace. In other words, the images of

all points in T under ϕ|πform a hyperplane of AG

( + 1, (q′)m−

). Furthermore, note that

T ⊆ π ⊆ Π+1 = τ+1. As B ∩ τ+1 is a union of + 1 concurrent, ϕ|τ+1-independent lines,

19

there has to exist a point Q+1 ∈ T∩B∩τ+1, since any union of +1 concurrent, independentlines of AG

(+ 1, (q′)m−

)meets any hyperplane in at least one point.

By recursively backtracking the observation obtained in step 3., we conclude that Q+1 liesin 〈Q, Q−1, . . . , Q1, P 〉, with Qj ∈ B ∩ τj (j ∈ {1, 2, . . . , + 1}). This implies that P ∈〈Q1, Q2, . . . , Q+1〉, as no point of {Q1, Q2, . . . , Q+1} can lie in the span of the others (else Fwould not be an (m− )-flower).

By the lemma above, we can find a relatively small point set that -saturates ‘most’ of the pointsof PG(N, q). We could end our quest right here and now, by copying smaller versions of similarpoint sets in the span of any petals of F . However, as this would dramatically increase thesize of the saturating set, we need to optimise the construction. Hence, to compensate for therestricted -saturating capabilities described by the lemma above, we construct a -saturatingset as a mix of several flowers. To do this, we fix following notation and definition.

Notation 7.2.2. Define the value λ := min{,N − }.

Definition 7.2.3. Let 0 < < N . For every i ∈ {1, . . . , λ}, define a map

⌈·⌋(i) : {+ 2− λ, . . . , + 1} → {+ 2− λ, . . . , + 1}

: j 7→ ⌈j⌋(i) :={j + i− 1 if j + i− 1 6 + 1,

+ 2− i otherwise.

As the map above could induce some confusion, we will give the reader an intuition of Construc-tion 7.2.4 (see below) before plunging into the technical details.As said before, the main construction will be built by making use of a mix of multiple flowers.These flowers will be stacked upon each other, forming a total of λ ‘layers’, in the sense that

• the ‘largest’ layer (layer i = 1) is an (N − )-flower with + 1 petals; the petals will benumbered 1, 2, . . . , + 1,

• within that layer, we consider an (N − − 1)-flower with + 1 petals (layer i = 2), suchthat each of the petals is contained in a unique petal of the layer above,

• within that layer, we consider an (N − − 2)-flower with + 1 petals (layer i = 3), suchthat each of the petals is contained in a unique petal of the layer above,

• ...

• the ‘smallest’ layer (layer i = λ) is an (N − − λ+ 1)-flower with + 1 petals, such thateach of the petals is contained in a unique petal of the layer above.

In this way, we obtain a large flower consisting of ‘layered’ petals numbered 1, 2, . . . , + 1.Inspired by Lemma 7.2.1, we now choose a set of concurrent, ϕ-independent q′-subgeometriesin certain layers of each of the petals. The number of such subgeometries will depend on thenumber of the layer (i) and the number of the petal (j). If j 6 + 1 − λ, we will choose jconcurrent, ϕ-independent q′-subgeometries in the top layer (i = 1) of petal j, and none inany of its other layers. If j > + 1− λ, then the value ⌈j⌋(i) equals the number of concurrent,ϕ-independent q′-subgeometries we will choose in layer i of petal j. To elaborate, if j > +1−λ,we will choose

• precisely ⌈j⌋(1) = j concurrent, ϕ-independent q′-subgeometries in the top layer of petalj,

• precisely ⌈j⌋(2) = j+1 concurrent, ϕ-independent q′-subgeometries in the next layer (i = 2)of petal j,

20

• ...

• precisely ⌈j⌋(+2−j) = + 1 concurrent, ϕ-independent q′-subgeometries in the next layer(i = + 2− j) of petal j,

• precisely ⌈j⌋(+3−j) = j − 1 concurrent, ϕ-independent q′-subgeometries in the next layer(i = + 3− j) of petal j,

• ...

• precisely ⌈j⌋(λ) = + 2 − λ concurrent, ϕ-independent q′-subgeometries in the bottomlayer (i = λ) of petal j.

We now formalise the intuitive construction above and hence introduce the main construction ofthis article. Be sure to keep Figure 1 at hand for a visualisation of an example case with threetwo-layered petals.

Construction 7.2.4. Let 0 < < N and let q = (q′)+1. Suppose {C1, . . . , Cλ} is a set ofq′-subgeometries and suppose {Σ1, . . . ,Σλ} is a set of subspaces of PG(N, q) with the followingproperties.

• For every i ∈ {1, . . . , λ}, Ci is an (N − − i)-dimensional q′-subgeometry such that Σi =〈Ci〉q.

• C1 ⊇ C2 ⊇ · · · ⊇ Cλ, hence Σ1 ⊇ Σ2 ⊇ · · · ⊇ Σλ.

Moreover, consider a set of flowers {F1, . . . ,Fλ} with the following properties.

• For every i ∈ {1, . . . , λ}, Fi :={τi1, . . . , τi(+1)

}is an (N − − i+1)-flower with pistil Σi.

• For every j ∈ {1, . . . , + 1}, τ1j ⊇ τ2j ⊇ · · · ⊇ τλj .

Now define, for every j ∈ {1, . . . , + 1},

Pj :=

⋃jk=1

(B(k)1j \ C1

)if j 6 + 1− λ,

⋃λi=1

⋃⌈j⌋(i)k=1

(B(k)ij \ Ci

)if j > + 1− λ,

where B(1)ij \ Ci,B(2)

ij \ Ci, . . . ,B(⌈j⌋(i))

ij \ Ci are ⌈j⌋(i) distinct, ϕ|τij -independent elements of LτijCi ,

all sharing a point Fij ∈ PτijCi \ τ(i+1)j (i ∈ {1, . . . , λ}, τ(λ+1)j := ∅).

Finally, define

P′1 :=

{⋃λi=2 (B′

i1 \ Ci) if q′ = 2,

∅ if q′ 6= 2,

with B′i1 \ Ci an element of Lτi1

Ci with the property that 〈B′i1〉q intersects B′

(i−1)1 only in Ci(i ∈ {2, . . . , λ}, B′

11 := B(1)11 ).

Lemma 7.2.5. Consider Construction 7.2.4. Then

|P′1| =

{(2λ−1 − 1) · 2N−−λ+1 if q′ = 2,

0 if q′ 6= 2.

Lemma 7.2.6. Consider Construction 7.2.4. Let j ∈ {1, . . . , + 1}. If j 6 + 1− λ, then

|Pj | = j(q′)N− − (j − 1).

If j > + 1− λ, then one can find a set Pj such that

|Pj | = j(q′)N− +

+1−j∑

k=1

(j − 1 + k)(q′)N−−k +

λ−1∑

k=+2−j

(− k)(q′)N−−k − λ(2− λ+ 1)

2.

21

PG(5, (q′)3

)PG

(5, (q′)3

)

Σ1

Σ2

C1

C2

τ11

F11

B(1)11

τ21

F21

τ12

F12

B(1)12 ∪ B

(2)12

τ22

B(1)

22∪B(2)

22∪B(3)

22

F22

τ13

F13

B(1)13 ∪ B

(2)13 ∪ B

(3)13

τ23B(1)23 ∪ B

(2)23

F23

Figure 1: A visualisation of Construction 7.2.4 in case N = 5 and = 2; we observe two stackedflowers, resulting in three two-layered petals. The petal τ11 has a number j = 1 not exceeding+ 1− λ = 1. The petals with number j = 2 correspond to ⌈2⌋(1) = 2 chosen q′-subgeometriesin the top layer and ⌈2⌋(2) = 3 chosen q′-subgeometries in the bottom layer (increasing). Thepetals with number j = 3 correspond to ⌈3⌋(1) = 3 chosen q′-subgeometries in the top layer and⌈3⌋(2) = 2 chosen q′-subgeometries in the bottom layer (decreasing).

22

Proof. If j 6 + 1 − λ, this result is easily obtained, as distinct elements of Lτ1jC1 can share at

most one point. Hence, assume j > + 1 − λ. To minimalise the size of Pj , we can always

choose B(1)ij to be a subspace of B(1)

(i−1)j , for every i ∈ {2, . . . , λ}. In this way, keeping the nature

of ⌈·⌋(·) in mind (see Definition 7.2.3), we obtain the following:

|Pj | = j(q′)N− − (j − 1)

+ j(q′)N−−1 − j

+ (j + 1)(q′)N−−2 − (j + 1)

......

+ (q′)N−−(+1−j) −

+ (j − 2)(q′)N−−(+2−j) − (j − 2)

+ (j − 3)(q′)N−−(+3−j) − (j − 3)

......

+ (+ 1− λ)(q′)N−−(λ−1) − (+ 1− λ).

Viewing the expression above as a polynomial in q′, the corresponding constant term equals

−((+ 1− λ) + · · · +

)=

(− λ)(+ 1− λ)

2− (+ 1)

2

= −λ(2− λ+ 1)

2.

Lemma 7.2.7. Consider Construction 7.2.4. Then we can find sets P1, . . . ,P+1 such that

+1∑

i=1

|Pi| =(+ 1)( + 2)

2(q′)N− +

λ−1∑

j=1

a(N, , j)(q′)N−−j − c(N, ),

with

a(N, , j) :=λ(2− λ+ 2j + 1)− j(3j + 1)

2and c(N, ) :=

(+ 1) + λ(λ− 1)(2 − λ+ 1)

2.

Proof. Let P1, . . . ,P+1 be sets of size equal to the values described in Lemma 7.2.6. Interpret∑+1i=1 |Pi| as a polynomial in q′ of degree N − ; let a(N, , j) be the coefficient corresponding

to (q′)N−−j (j ∈ {0, 1, . . . , N − − 1}) and let −c(N, ) be the constant term.

23

It is clear that a(N, , 0) =∑+1

i=1 i = (+1)(+2)2 . Furthermore, we can deduce that

a(N, , 1) = (+ 2− λ) + (+ 3− λ) + · · ·+ ()︸ ︷︷ ︸arising from P+2−λ,P+3−λ, ... ,P

+ (− 1)︸ ︷︷ ︸arising from P+1

=λ(2− λ+ 3

)− 4

2,

a(N, , 2) = (+ 3− λ) + (+ 4− λ) + · · ·+ ()︸ ︷︷ ︸arising from P+2−λ,P+3−λ, ... ,P−1

+ 2(− 2)︸ ︷︷ ︸arising from P and P+1

=λ(2− λ+ 5

)− 14

2,

...

a(N, , j) = (+ j + 1− λ) + (+ j + 2− λ) + · · ·+ ()︸ ︷︷ ︸arising from P+2−λ,P+3−λ, ... ,P+1−j

+ j(− j)︸ ︷︷ ︸arising from P+2−j , ... ,P+1

=λ(2− λ+ 2j + 1)− j(3j + 1)

2,

...

a(N, , λ − 1) = ()︸︷︷︸arising from P+2−λ

+ (λ− 1)( + 1− λ)︸ ︷︷ ︸arising from P+3−λ, ... ,P+1

= λ− λ2 + 2λ− 1,

a(N, , λ) = · · · = a(N, ,N − − 1) = 0,

−c(N, ) = −1− 2− 3− · · · − (− λ)︸ ︷︷ ︸arising from P2,P3, ... ,P+1−λ

+ λ

(−λ(2− λ+ 1)

2

)

︸ ︷︷ ︸arising from P+2−λ,P+3−λ, ... ,P+1

= −(+ 1) + λ(λ− 1)(2 − λ+ 1)

2.

Lemma 7.2.8. Consider Construction 7.2.4. Then the point set

B(N,) := P′1 ∪

+1⋃

j=1

Pj

-saturates all points of PG(N, q) \ Σ1.

Proof. Let P be an arbitrary point of PG(N, q) \ Σ1 and let

µ := min{|F| : F ⊆ F1, P ∈

⟨τ : τ ∈ F

⟩}∈ {1, 2, . . . , + 1}.

Hence, there exists an (N − )-flower F ′1 := {τ ′11, τ ′12, . . . , τ ′1µ} ⊆ F1 with pistil Σ1 such that P

lies in the span of all µ petals of F ′1, but does not lie in the span of any µ− 1 petals of F ′

1.For each petal τ ′1j , j ∈ {2, 3, . . . , µ − 1}, only the points of B(N,) in the top layer (i = 1) ofτ ′1j will be used to prove point saturation. As a consequence, we can assume without loss ofgenerality that τ ′1j = τ1j for every j ∈ {2, 3, . . . , µ − 1}. If q′ > 2, the same can be said aboutpetal τ ′11. If q

′ = 2, however, two possibilities can occur:

i either there exist at least two (N −)-dimensional q′-subgeometries B1 and B2 in τ ′11, bothcontaining C1 and a point F ∈ τ ′11 \ Σ1, such that (B1 ∪ B2) \ C1 ⊆ B(N,), or

ii τ ′11 = τ11.

24

Note that for both possibilities i and ii, there exists one (N − )-dimensional q′-subgeometryB in τ ′11 containing C1 such that B \ C1 ⊆ B(N,); this is the only property needed of petal τ ′11in Case 1, Case 2 and Case 3 (step 1. and 2.) below. Only in Case 3 (step 3.) a distinctionbetween possibility i and ii has to be made. In light of this, we will, for now, assume thatτ ′11 = τ11, and will remove this assumption in the third step of Case 3. Finally, we may assumethat τ ′1µ ∈ {τ1µ, τ1(µ+1), . . . , τ1(+1)}, hence there has to exist a j′ > µ such that τ ′1µ = τ1j′ . Inconclusion, assume that F ′

1 = {τ11, τ12, . . . , τ1(µ−1), τ1j′} for a certain j′ > µ.If µ = + 1, the proof follows immediately due to Lemma 7.2.1. We consider three cases,depending on the other possible values of µ.

Case 1: µ = 1.

In this case, P is contained in τ11, which is an (N−)-dimensional subspace containing B(1)11 \C1 ⊆

B(N,). By Lemma 7.1.3, there exists an r-subspace π of B(1)11 , 0 < r 6 , such that 〈π〉q contains

P . As P /∈ Σ1, π \ Σ1 has to be isomorphic to AG(r, q′), hence we can easily find r + 1 points

in π \ Σ1 ⊆ B(1)11 \ Σ1 spanning 〈π〉q.

Case 2: 1 < µ 6 + 1− λ.

Note that the occurrence of this case implies that λ = N − .

We can choose N − + 1 points of B(1)1j′ \ C1 spanning the subspace τ1j′ ⊇ Σ1, and one point in

each set B(1)11 \ C1, . . . ,B(1)

1(µ−1) \ C1. These choices result in a total of (N − + 1) + (µ − 1) 6

(N − + 1) + (− λ) = + 1 points spanning⟨τ11, τ12, . . . , τ1(µ−1), τ1j′

⟩∋ P .

Case 3: + 1− λ < µ 6 .

Consider the following series of steps.

1. For every j ∈ {1, 2, . . . , µ − 1} ∪ {j′}, define τ2j := 〈Σ2, F1j〉 and consider, for every

k ∈ {1, 2, . . . , j}, the (N − − 1)-dimensional q′-subgeometry A(k)2j := B(k)

1j ∩ 〈Σ2, F1j〉. Inthis way, we obtain, for every j ∈ {1, 2, . . . , µ− 1}∪{j′}, a union A(1)

2j ∪A(2)2j ∪ · · · ∪A(j)

2j ofj distinct, ϕ|τ2j -independent (N − − 1)-dimensional q′-subgeometries in an (N − − 1)-dimensional subspace τ2j of τ1j, each containing C2 and sharing the point F1j .

2. Consider the union B(1)2j′ ∪ B(2)

2j′ ∪ · · · ∪ B(⌈j′⌋(2))

2j′ of ⌈j′⌋(2) distinct, ϕ|τ2j′ -independent (N −− 1)-dimensional q′-subgeometries in the (N − − 1)-dimensional subspace τ2j′ 6= τ2j′ ofτ1j′ , each containing C2 and sharing the point F2j′ . It is clear that τ2j′ and τ2j′ span thespace τ1j′ , as these are distinct hyperplanes of the latter space.

3. As described at the start of this proof, we have to remove the assumption that τ ′11 = τ11.

Note that⟨τ22, τ23, . . . , τ2(µ−1), τ2j′

⟩and

⟨τ22, τ23, . . . , τ2(µ−1), τ2j′

⟩both span hyperplanes

of⟨τ ′11, τ12, . . . , τ1(µ−1), τ1j′

⟩that do not contain τ ′11, hence each of these hyperplanes in-

tersect τ ′11 in an (N − − 1)-subspace τ21 and τ21, respectively, both containing C2.The goal is to find an (N−−1)-flower with µ+1 petals such that the jth petal contains jconcurrent, ϕ-independent (N −−1)-dimensional q′-subgeometries contained in B(N,)∪C2, with the additional property that P lies in the span of these petals, but not in the spanof any µ petals. It is clear that, if we can find an (N − − 1)-subspace τ21 /∈ {τ21, τ21}in τ ′11, not lying in Σ1 and containing an (N − − 1)-dimensional q′-subgeometry B ⊇ C2,B\C2 ⊆ B(N,), then {τ21, τ22, . . . , τ2(µ−1), τ2j′ , τ2j′} is the (N−−1)-flower we are lookingfor.

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• If q′ > 2, then there exists an (N − − 1)-dimensional subspace of B(1)11 spanning an

(N − − 1)-space τ21 that contains Σ2, but is not equal to Σ1, τ21 or τ21.

• If q′ = 2, we distinguish the two possibilities described at the start of the proof:

i suppose there exist at least two (N−)-dimensional q′-subgeometries B1 and B2 inτ ′11, both containing C1 and a point F ∈ τ ′11\Σ1, such that (B1 ∪ B2)\C1 ⊆ B(N,).By Lemma 7.1.2, we find at least three (N − − 1)-spaces through Σ2, not lyingin Σ1, that intersect either B1 or B2 in an (N−−1)-dimensional q′-subgeometry.Hence, one of these three (N − − 1)-spaces τ12 cannot be equal to τ21 or τ21.

ii if τ ′11 = τ11, then, by the definition of the setP′1, we can always find an (N−−1)-

dimensional subspace τ21 with the desired properties.

Intuitively, the steps above split the initial (N − )-flower with µ petals into an (N − − 1)-flower with µ + 1 petals. For this new flower, the property that P is contained in the spanof all of its petals, but not in the span of any fewer number of petals, still holds. We executethe steps above a total of + 1 − µ times, leaving us with an (N − 2 + µ − 1)-flower F ′

+2−µ

with +1 petals. Note that this is always possible, as by the assumption corresponding to thiscase, + 1 − µ 6 λ − 1, which means that, in each step, one can always choose smaller petalscontaining subgeometries (see Construction 7.2.4) which fulfil the desired conditions.Moreover, for each j ∈ {1, . . . , + 1}, there must exist a petal in F ′

+2−µ with j concurrent, ϕ-independent (N −2+µ−1)-dimensional q′-subgeometries contained in B(N,) ∪C1. Indeed, letLi be the list of numbers of concurrent, ϕ-independent (N −− i)-dimensional q′-subgeometrieswe can find in the respective petals of the flower we obtain after going through the steps i times(i ∈ {0, 1, . . . , + 1 − µ}). Then, by considering the nature of the maps ⌈·⌋(·) (see Definition7.2.3), we get

L0 = (1, 2, . . . , µ− 1, j′),

L1 = (1, 2, . . . , µ− 1, j′, j′ + 1),

L2 = (1, 2, . . . , µ− 1, j′, j′ + 1, j′ + 2),

...

L+1−j′ = (1, 2, . . . , µ− 1, j′, j′ + 1, . . . , + 1),

L−j′ = (1, 2, . . . , µ− 1, j′ − 1, j′, j′ + 1, . . . , + 1),

...

L+1−µ = (1, 2, . . . , µ− 1, µ, µ + 1, . . . , j′ − 1, j′, j′ + 1, . . . , + 1).

Hence, Lemma 7.2.1 finishes the proof.

Theorem 7.2.9. Let 0 < < N and let q = (q′)+1 for any prime power q′. Then

sq(N, ) 6

k(N,)∑

i=1

((+ 1)( + 2)

2(q′)N+1−i(+1)

)+

k(N,)−1∑

i=1

−1∑

j=1

a(, j)(q′)N+1−i(+1)−j

+

ℓ(N,)−1∑

j=1

a(N, , j)(q′)ℓ(N,)−j − c(N, )− c(N, )

+ δq′=2 ·

(2−1 − 1) ·

k(N,)−1∑

i=1

(2N−+2−i(+1)

)+ 2ℓ(N,) − 2

,

with

26

• k(N, ) :=⌈N−+1

⌉,

• ℓ(N, ) := N + 1− k(N, ) · (+ 1) =(N (mod + 1)

)+ 1,

• a(, j) := (+2j+1)−j(3j+1)2 6

(2+1)3 ,

• a(N, , j) :=ℓ(N,)

(2−ℓ(N,)+2j+1

)−j(3j+1)

2 6 a(, j),

• c(N, ) :=(k(N, ) − 1

)2(+1)2 > 0,

• c(N, ) :=(+1)+ℓ(N,)

(ℓ(N,)−1

)(2−ℓ(N,)+1

)2 > 0,

• δq′=2 :=

{1 if q′ = 2,

0 if q′ 6= 2.

Proof. First, let us assume N + 1 is a multiple of + 1. Then k(N, ) = N−+1 and ℓ(N, ) =

+ 1. Moreover, if we interpret a(, j) and a(N, , j) as quadratic polynomials in j (withj ∈ {1, . . . , − 1} and j ∈ {1, . . . , }, respectively), then one can check that these polynomialsreach their minimum values if j ∈ {1, − 1} and j ∈ {1, }, respectively. Hence, as > 1, wehave

0 6 2− 2 6 min{a(, 1), a(, − 1)

}6 a(, j) (11)

for every j ∈ {1, . . . , − 1}, and

0 6 6 min{a(N, , 1), a(N, , )

}6 a(N, , j), (12)

for every j ∈ {1, . . . , }. Furthermore, the fact that N + 1 is a multiple of + 1 togetherwith < N implies that 2 + 1 6 N , hence we obtain c(N, ) 6 1

2( + 1)(N − 2 − 1) andc(N, ) = 1

2( + 1)2. Also note that k(N, ) > 1. If we denote with B(N, ) the upper bounddescribed in this theorem, we can combine these results, together with (11) and (12), to concludethat

B(N, ) >

k(N,)∑

i=1

((+ 1)( + 2)

2(q′)N+1−i(+1)

)− 1

2(+ 1)(N − 2− 1)− 1

2(+ 1)2 + δq′=2

=

k(N,)∑

i=1

((+ 1)(q′)N+1−i(+1)

)

+

k(N,)∑

i=1

(1

2(+ 1)(q′)N+1−i(+1)

)− 1

2(+ 1)(N − ) + δq′=2

>

k(N,)∑

i=1

((+ 1)(q′)N+1−i(+1)

)+

1

2(+ 1)(q′)N− − 1

2(+ 1)(N − ) + δq′=2

= (+ 1)(qk(N,) + qk(N,)−1 + · · · + q

)+

1

2(+ 1)

((q′)N− − (N − )

)+ δq′=2

> (+ 1)(qk(N,) + qk(N,)−1 + · · · + q

)+ (+ 1) > sq(N, ).

The latter inequality follows from Corollary 5.1.2.As a result, we can assume that N + 1 is not a multiple of + 1.By Lemma 7.2.8, we can choose a point set B(N,) in PG(N, q) (described in Construction7.2.4) which -saturates all points of PG(N, q), except for the points contained in a certain(N − − 1)-subspace Σ.

27

If N − − 1 6 , then N − − 1 < , as else N +1 would be a multiple of +1. Hence, in thiscase, all points of Σ are -saturated by B(N,) as well, as we can simply choose + 1 points inP1 that span the subspace τ11 ⊇ Σ.If N − − 1 > , then, by Lemma 7.2.8, we can choose a point set B(N−(+1),) in Σ which-saturates all points of Σ, except for the points contained in a certain

(N − 2(+1)

)-subspace

of Σ. We can repeat this process to obtain a union

B(N,) ∪B(N−(+1),) ∪ · · · ∪B(N−(k(N,)−1)(+1),)

of k(N, ) point sets that -saturates all points of PG(N, q).For each i ∈ {1, 2, . . . , k(N, )}, the size of the point set B(N−(i−1)(+1),) can be calculated usingLemma 7.2.5 and Lemma 7.2.7, where every instance of N has to be replaced by N−(i−1)(+1),hence every instance of λ has to be replaced by λi := min{,N − (i− 1)(+ 1)− }.

• If i ∈ {1, 2, . . . , k(N, )− 1}, then < N − (i− 1)(+ 1)− , which implies that λi = .

• If i = k(N, ), then > N −(k(N, ) − 1

)( + 1) − (keeping in mind that N + 1 is no

multiple of + 1), which implies that λk(N,) = ℓ(N, ).

Finally, we claim that a(N, , j) 6 a(, j) 6 (2+1)3 , for all j ∈ {1, . . . , }. Indeed, for the first

inequality, one can interpret ℓ(N,)(2−ℓ(N,))2 as a quadratic polynomial in ℓ(N, ), which reaches

its maximum value if ℓ(N, ) = . For the second inequality, one can interpret a(, j) as aquadratic polynomial in j, which reaches its maximum value if j = 2−1

6 . However, the latter isnever an integer. Hence, one can conclude that

a(, j) 6 max

{a

(,

2− 1

6− 1

6

), a

(,

2− 1

6+

1

6

)}=

(2+ 1)

3,

for all j ∈ {1, . . . , }.

As the upper bound presented in Theorem 7.2.9 is not easy to work with in practice, a simplifiedupper bound is desired. If is large enough, the upper bound of Theorem 7.2.9 simplifiesconsiderably. More precisely, if > N−1

2, then k(N, ) = 1 and ℓ(N, ) = N − , hence the

bound of Theorem 7.2.9 becomes the following:

sq(N, ) 6(+ 1)(+ 2)

2(q′)N− +

N−−1∑

j=1

a(N, , j)(q′)N−−j − c(N, ) + δq′=2 ·(2N− − 2

).

In case > 1, one can deduce from Theorem 7.2.9 the following easy-to-read but slightly weakerbound, which we present as our main result.

Theorem 7.2.10. Let 1 < < N and let q = (q′)+1 for any prime power q′. Then

sq(N, ) 6(+ 1)( + 2)

2(q′)N− + (+ 1)

(q′)N− − 1

q′ − 1.

Translating the result above in coding theoretical terminology (see Subsection 4.1), one obtainsthe following.

Corollary 7.2.11. Let 2 < R < r and let q = (q′)R for any prime power q′. Then

ℓq(r,R) 6R(R+ 1)

2(q′)r−R + (R− 1)R

(q′)r−R − 1

q′ − 1.

For any infinite family of covering codes of length equal to the upper bound above, the followingholds for its asymptotic covering density:

µq(R) <

((R− 1)R

)R

R!

(1 +

1

q′+ · · ·+ 1

(q′)R−1

)R

<

(e(R − 1)

q − 1

q − (q′)R−1

)R

,

with e ≈ 2.718... being Euler’s number.

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Open Problem 7.2.12. Given that q = (q′)+1 = (q′)R, improve the leading coefficient(+1)(+2)

2 of Theorem 7.2.9 and 7.2.10 (respectively the leading coefficient R(R+1)2 of Corollary

7.2.11) to one that is linear in (respectively linear in R).

Acknowledgements. First of all, I would like to thank Fernanda Pambianco for her opinion onthese results, as well as her guidance through the many literary works on this topic. Secondly,I would like to express my appreciation for the quick responses and helpful replies of MaartenDe Boeck and Geertrui Van de Voorde on the topic of point-line geometries and linear represen-tations. A special thanks goes towards Stefaan De Winter for his incredibly elegant sketch onhow to construct an isomorphism between Y (,m, q′) and X(,m, q′) using field reduction (seeSubsection 6.4), which motivated me to construct the direct isomorphism described in Section6. Finally, I would like to thank my supervisor Leo Storme and colleague Jozefien D’haeseleerfor their time and effort in proofreading this work, as well as the anonymous referees for theirinsightful corrections and remarks.

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Author’s address:

Lins DenauxGhent UniversityDepartment of Mathematics: Analysis, Logic and Discrete MathematicsKrijgslaan 281 – Building S89000 GhentBELGIUMe-mail : lins.denaux@ugent.be

website: https://users.ugent.be/~ldnaux

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