Maximum Scattered Linear Sets in Finite Projective Spaces

• Maximum scattered linear sets are F₍q₎-linear sets in projective spaces where every point has weight 1, achieving the highest possible rank and a specific size formula.
• They bridge algebraic and geometric frameworks, providing concrete links to MRD codes, finite semifields, blocking sets, and translation structures.
• Recent studies reveal a rigid classification in low dimensions (n = 4, 5) alongside diverse constructions in larger even-dimensional regimes.

Maximum scattered linear sets are scattered $\mathbb{F}_q$-linear sets of highest possible rank in a projective space $\mathrm{PG}(r-1,q^n)$. If $U$ is an $\mathbb{F}_q$-subspace of dimension $k$ of an $r$-dimensional $\mathbb{F}_{q^n}$-vector space $V$, the associated linear set is $$L_U=\{(u)_{\mathbb{F}_{q^n}}:u\in U\setminus\{0\}\}$$, its rank is $\dim_{\mathbb{F}_q}U$, and it is scattered when every point has weight $\mathrm{PG}(r-1,q^n)$0, equivalently when $\mathrm{PG}(r-1,q^n)$1. In $\mathrm{PG}(r-1,q^n)$2 the maximum possible rank is $\mathrm{PG}(r-1,q^n)$3, so a maximum scattered linear set has rank $\mathrm{PG}(r-1,q^n)$4 and size $\mathrm{PG}(r-1,q^n)$5. These objects are central in finite geometry because they are tightly connected with MRD codes, finite semifields, blocking sets, translation structures, spreads, ovoids, and flocks (Csajbók et al., 2017, Lia et al., 31 Jul 2025, Longobardi, 2024).

1. Definitions and extremal bounds

Let $\mathrm{PG}(r-1,q^n)$6 be the projective space over $\mathrm{PG}(r-1,q^n)$7. If $\mathrm{PG}(r-1,q^n)$8 is an $\mathrm{PG}(r-1,q^n)$9-subspace of $U$0, the point weight of $U$1 in $U$2 is

$U$3

A linear set is scattered precisely when all points have weight $U$4. For scattered linear sets, Blokhuis and Lavrauw proved the rank bound

$U$5

when $U$6 is even. Bartoli, Giulietti, Marino, and Polverino proved that this bound is sharp for all $U$7 with $U$8 even: for every such triple there exist scattered $U$9-linear sets of rank $\mathbb{F}_q$0, and these are called maximum scattered linear sets (Csajbók et al., 2017).

On the projective line $\mathbb{F}_q$1, the theory simplifies but becomes more rigid. Every scattered linear set has rank at most $\mathbb{F}_q$2, and a maximum scattered linear set has rank $\mathbb{F}_q$3. Its size is

$\mathbb{F}_q$4

For $\mathbb{F}_q$5, maximum scattered linear sets are therefore extremal both in rank and in cardinality, and the corresponding $\mathbb{F}_q$6-subspaces of $\mathbb{F}_q$7 are often called maximum scattered subspaces (Csajbók et al., 2017, Longobardi, 2024).

2. Algebraic and geometric models

In $\mathbb{F}_q$8, every rank-$\mathbb{F}_q$9 linear set can be written, after a projectivity, in the form

$k$0

where $k$1 is an $k$2-linearized polynomial. The scattered condition is

$k$3

Such $k$4 are called scattered polynomials. The correspondence $k$5 is exact: $k$6 is scattered if and only if $k$7 is a maximum scattered linear set, and this is also equivalent to the associated code

$k$8

being an MRD code with parameters $k$9 (Lia et al., 31 Jul 2025, Longobardi, 2024).

A complementary geometric model is given by projections of canonical subgeometries. Lunardon and Polverino proved that every $r$0-linear set of rank $r$1 in $r$2 is the projection of a canonical $r$3-subgeometry $r$4 of $r$5 from a complementary subspace $r$6. In the case of a maximum scattered linear set of $r$7, one works in $r$8 with a plane $r$9 projecting $\mathbb{F}_{q^n}$0 onto a line. Scatteredness is detected by the position of $\mathbb{F}_{q^n}$1: the projection is scattered if and only if every point of $\mathbb{F}_{q^n}$2 has rank $\mathbb{F}_{q^n}$3, equivalently $\mathbb{F}_{q^n}$4 is external to the secant variety to $\mathbb{F}_{q^n}$5 (Lia et al., 31 Jul 2025).

This projection formalism underlies several classification results. In $\mathbb{F}_{q^n}$6, for a projecting configuration $\mathbb{F}_{q^n}$7 and a generator $\mathbb{F}_{q^n}$8 of $\mathbb{F}_{q^n}$9, the intersections

$V$0

encode the type of the linear set. Their dimensions and their ranks with respect to $V$1 sharply constrain whether the projection is of pseudoregulus type, of LP type, or a possible new type (Lia et al., 31 Jul 2025).

3. Principal families and equivalence

Several explicit families of maximum scattered linear sets on $V$2 are known. The principal infinite families are summarized below.

Family	Polynomial form	Parameter conditions
Pseudoregulus type	$V$3	$V$4
LP type	$V$5	$V$6
Longobardi–Zanella family	$V$7	$V$8, with conditions on $V$9
Large family in even dimension	$$L_U=\{(u)_{\mathbb{F}_{q^n}}:u\in U\setminus\{0\}\}$$0	$$L_U=\{(u)_{\mathbb{F}_{q^n}}:u\in U\setminus\{0\}\}$$1, $$L_U=\{(u)_{\mathbb{F}_{q^n}}:u\in U\setminus\{0\}\}$$2 odd, $$L_U=\{(u)_{\mathbb{F}_{q^n}}:u\in U\setminus\{0\}\}$$3

The pseudoregulus family is the classical monomial family. The LP family, due to Lunardon and Polverino, is the first non-pseudoregulus infinite family and, for $$L_U=\{(u)_{\mathbb{F}_{q^n}}:u\in U\setminus\{0\}\}$$4, includes the forms

$$L_U=\{(u)_{\mathbb{F}_{q^n}}:u\in U\setminus\{0\}\}$$5

For $$L_U=\{(u)_{\mathbb{F}_{q^n}}:u\in U\setminus\{0\}\}$$6, the two LP families are those with $$L_U=\{(u)_{\mathbb{F}_{q^n}}:u\in U\setminus\{0\}\}$$7 (Csajbók et al., 2017, Lia et al., 31 Jul 2025).

Longobardi and Zanella constructed a further infinite family for even $$L_U=\{(u)_{\mathbb{F}_{q^n}}:u\in U\setminus\{0\}\}$$8, starting from

$$L_U=\{(u)_{\mathbb{F}_{q^n}}:u\in U\setminus\{0\}\}$$9

and proved that $\dim_{\mathbb{F}_q}U$0 is scattered exactly when either $\dim_{\mathbb{F}_q}U$1 is even and $\dim_{\mathbb{F}_q}U$2, or $\dim_{\mathbb{F}_q}U$3 is odd, $\dim_{\mathbb{F}_q}U$4, and $\dim_{\mathbb{F}_q}U$5. This yields new maximum scattered linear sets in $\dim_{\mathbb{F}_q}U$6 for $\dim_{\mathbb{F}_q}U$7, and new MRD codes with parameters $\dim_{\mathbb{F}_q}U$8 (Longobardi et al., 2020).

A different large family for even $\dim_{\mathbb{F}_q}U$9 and odd $\mathrm{PG}(r-1,q^n)$00 is given by the polynomials $\mathrm{PG}(r-1,q^n)$01, defined for $\mathrm{PG}(r-1,q^n)$02 with $\mathrm{PG}(r-1,q^n)$03. These produce many pairwise inequivalent maximum scattered linear sets. For fixed $\mathrm{PG}(r-1,q^n)$04 and $\mathrm{PG}(r-1,q^n)$05, the number $\mathrm{PG}(r-1,q^n)$06 of $\mathrm{PG}(r-1,q^n)$07-inequivalent members satisfies

$\mathrm{PG}(r-1,q^n)$08

This markedly enlarges the known landscape for even $\mathrm{PG}(r-1,q^n)$09 (Longobardi et al., 2021).

Equivalence is usually taken under $\mathrm{PG}(r-1,q^n)$10, $\mathrm{PG}(r-1,q^n)$11, or $\mathrm{PG}(r-1,q^n)$12. For Lunardon–Polverino linear sets, equivalence is controlled by norm data of the parameter $\mathrm{PG}(r-1,q^n)$13, and their automorphism groups were determined explicitly in terms of diagonal and, in special cases, off-diagonal semilinear maps (Tang et al., 2022). For two further families in $\mathrm{PG}(r-1,q^n)$14 and $\mathrm{PG}(r-1,q^n)$15, equivalence was likewise reduced to norm conditions over intermediate subfields, and automorphism groups were computed (Tang et al., 2022).

4. Classification for small $\mathrm{PG}(r-1,q^n)$16

For $\mathrm{PG}(r-1,q^n)$17, the projective-line classification is very tight. In $\mathrm{PG}(r-1,q^n)$18, Csajbók and Zanella proved that every maximum scattered $\mathrm{PG}(r-1,q^n)$19-linear set is projectively equivalent to

$\mathrm{PG}(r-1,q^n)$20

with $\mathrm{PG}(r-1,q^n)$21. Hence every maximum scattered linear set in $\mathrm{PG}(r-1,q^n)$22 is either of pseudoregulus type ($\mathrm{PG}(r-1,q^n)$23) or of Lunardon–Polverino type ($\mathrm{PG}(r-1,q^n)$24), and there are exactly $\mathrm{PG}(r-1,q^n)$25 projective equivalence classes (Csajbók et al., 2017).

The case $\mathrm{PG}(r-1,q^n)$26 is the first open line case where a full classification appears plausible but is not yet complete. Montanucci and Zanella studied the two-parameter family

$\mathrm{PG}(r-1,q^n)$27

proved strong necessary conditions for scatteredness, and showed that for $\mathrm{PG}(r-1,q^n)$28 every maximum scattered member of this family is equivalent to a known Sheekey example. In particular, if $\mathrm{PG}(r-1,q^n)$29 and $\mathrm{PG}(r-1,q^n)$30 is maximum scattered, then necessarily

$\mathrm{PG}(r-1,q^n)$31

(Montanucci et al., 2019).

A more structural advance was obtained in the 2025 work on $\mathrm{PG}(r-1,q^n)$32. Every maximum scattered linear set is the projection of $\mathrm{PG}(r-1,q^n)$33 from a plane $\mathrm{PG}(r-1,q^n)$34 external to the secant variety to $\mathrm{PG}(r-1,q^n)$35. If $\mathrm{PG}(r-1,q^n)$36 and $\mathrm{PG}(r-1,q^n)$37 are not both points, the linear set is of pseudoregulus type. If they are points and at least one of them has rank $\mathrm{PG}(r-1,q^n)$38, the linear set is of LP type. Hence any hypothetical new type must satisfy

$\mathrm{PG}(r-1,q^n)$39

In that case, up to equivalence, the linear set must belong to one of two explicit families: $\mathrm{PG}(r-1,q^n)$40 with $\mathrm{PG}(r-1,q^n)$41, $\mathrm{PG}(r-1,q^n)$42, $\mathrm{PG}(r-1,q^n)$43, or

$\mathrm{PG}(r-1,q^n)$44

where

$\mathrm{PG}(r-1,q^n)$45

An exhaustive analysis by computer shows that for $\mathrm{PG}(r-1,q^n)$46, none of these gives a new maximum scattered linear set: either the set is not scattered, or it is equivalent to pseudoregulus or LP type (Lia et al., 31 Jul 2025).

Another rigidity result concerns exceptional scattered polynomials. For exceptional scattered monic polynomials of index $\mathrm{PG}(r-1,q^n)$47, the only example for $\mathrm{PG}(r-1,q^n)$48 is $\mathrm{PG}(r-1,q^n)$49. For index $\mathrm{PG}(r-1,q^n)$50, the only possibilities are $\mathrm{PG}(r-1,q^n)$51 and

$\mathrm{PG}(r-1,q^n)$52

This strongly limits the exceptional sources of maximum scattered linear sets on projective lines (Bartoli et al., 2017).

5. Higher-dimensional existence and related structures

Beyond the line, maximum scattered linear sets exist in all projective spaces $\mathrm{PG}(r-1,q^n)$53 with $\mathrm{PG}(r-1,q^n)$54 even, and the extremal rank is always $\mathrm{PG}(r-1,q^n)$55 (Csajbók et al., 2017). Explicit constructions were given in $\mathrm{PG}(r-1,q^n)$56 via subspaces of the form

$\mathrm{PG}(r-1,q^n)$57

leading to maximum scattered linear sets of rank $\mathrm{PG}(r-1,q^n)$58, and then lifted to higher dimension by direct sums (Csajbók et al., 2017, Bartoli et al., 2015).

These higher-dimensional objects have several geometric and coding-theoretic avatars. Bartoli, Giulietti, Marino, and Polverino showed that every maximum scattered $\mathrm{PG}(r-1,q^n)$59-linear set $\mathrm{PG}(r-1,q^n)$60 yields an MRD code

$\mathrm{PG}(r-1,q^n)$61

with parameters

$\mathrm{PG}(r-1,q^n)$62

and $\mathrm{PG}(r-1,q^n)$63 is MRD if and only if $\mathrm{PG}(r-1,q^n)$64 is maximum scattered (Csajbók et al., 2017).

For $\mathrm{PG}(r-1,q^n)$65, maximum scattered $\mathrm{PG}(r-1,q^n)$66-linear sets correspond to translation caps. In particular, maximum scattered linear sets in $\mathrm{PG}(r-1,q^n)$67 with $\mathrm{PG}(r-1,q^n)$68 odd yield maximal translation caps in $\mathrm{PG}(r-1,q^n)$69, and the doubling construction gives complete caps in $\mathrm{PG}(r-1,q^n)$70 of size $\mathrm{PG}(r-1,q^n)$71. This resolves, for even square order, the long-standing problem of whether the theoretical lower bound for the size of a complete cap is substantially sharp (Bartoli et al., 2015).

A further development concerns pairwise disjoint maximum scattered linear sets. Families of disjoint maximum scattered linear sets correspond to maximum $\mathrm{PG}(r-1,q^n)$72-designs and to MSRD codes in the sum-rank metric. For any integers $\mathrm{PG}(r-1,q^n)$73 such that $\mathrm{PG}(r-1,q^n)$74 is even, and any $\mathrm{PG}(r-1,q^n)$75, there exist pairwise disjoint maximum scattered $\mathrm{PG}(r-1,q^n)$76-linear sets in $\mathrm{PG}(r-1,q^n)$77, hence maximum $\mathrm{PG}(r-1,q^n)$78-design systems with $\mathrm{PG}(r-1,q^n)$79 components (Santonastaso et al., 2023).

6. Present landscape

The current picture combines rigidity in small dimensions with abundance in some larger even dimensions. For $\mathrm{PG}(r-1,q^n)$80, the classification is complete: only pseudoregulus and LP type occur (Csajbók et al., 2017). For $\mathrm{PG}(r-1,q^n)$81, the known structure theorem reduces any possible new type to the two explicit families $\mathrm{PG}(r-1,q^n)$82 and $\mathrm{PG}(r-1,q^n)$83, and no new example exists for $\mathrm{PG}(r-1,q^n)$84 (Lia et al., 31 Jul 2025). This suggests a highly constrained geometry in $\mathrm{PG}(r-1,q^n)$85, although the absolute non-existence of new types for all $\mathrm{PG}(r-1,q^n)$86 is still open.

For larger even $\mathrm{PG}(r-1,q^n)$87, the situation is richer. The Longobardi–Zanella construction and the $\mathrm{PG}(r-1,q^n)$88-family show that new maximum scattered linear sets and new MRD codes do occur in dimensions $\mathrm{PG}(r-1,q^n)$89, and the number of inequivalent examples can be very large (Longobardi et al., 2020, Longobardi et al., 2021). A plausible implication is that the classification problem changes character between the small cases $\mathrm{PG}(r-1,q^n)$90 and the broader even-dimensional regime.

At the same time, equivalence and automorphism problems remain fundamental. For Lunardon–Polverino linear sets, automorphism groups, equivalence, and asymptotic counts of inequivalent members are known explicitly (Tang et al., 2022). For two other families in $\mathrm{PG}(r-1,q^n)$91 and $\mathrm{PG}(r-1,q^n)$92, equivalence and automorphism groups are also known (Tang et al., 2022). These results reinforce a general theme: maximum scattered linear sets are best understood not only through existence and scatteredness, but also through their orbit structure under $\mathrm{PG}(r-1,q^n)$93, their stabilizers, and the invariants they induce on associated MRD codes and semifields.