Regular Fat Linear Sets in Finite Geometry

• Regular fat linear sets are algebraically defined subsets in projective geometry with prescribed weight distributions that generalize scattered sets and clubs.
• They are constructed using q-linearized polynomials and subspace methods to ensure exactly r higher-weight points amidst a majority of weight-one points.
• These sets underpin applications in finite geometry and coding theory, particularly in designing three‐weight rank-metric codes and solving interpolation problems.

A regular fat linear set is a structured, algebraically defined subset of points in finite or projective geometry generalizing the classical notion of scattered linear sets by controlling the distribution of higher-weight points. The concept, which unifies a spectrum of geometric and algebraic constructions (including clubs, scattered sets, complementary-weight sets), serves as a fundamental tool in the interplay between algebraic combinatorics, finite geometry, interpolation theory, and the theory of few-weight codes.

1. Formal Definition and Weight Distribution

Let $U$ be an $\mathbb{F}_q$-subspace of dimension $\rho$ in the $k$-dimensional vector space $\mathbb{F}_{q^n}^k$. The associated $\mathbb{F}_q$-linear set in projective space $\mathrm{PG}(k-1, q^n)$ is

$$L_U = \{\langle v \rangle_{q^n} : v \in U \setminus \{0\} \}.$$

For $P = \langle v \rangle_{q^n}$, the $\mathbb{F}_q$-weight is given by

$$w_{L_U}(P) = \dim_q(U \cap \langle v \rangle_{q^n}).$$

An \textbf{$(r, i)$-regular fat linear set} is an $\mathbb{F}_q$-linear set $L_U$ where:

• Exactly $r$ points have weight $i \geq 2$,
• All other points have weight $1$,
• There is at least one point of weight $1$.

For $k = 2$, $U$ is typically described via a $q$-linearized polynomial $f \in \mathbb{F}_{q^n}[X]$, and

$$L_f = \{\langle (x : f(x)) \rangle_{q^n} : x \in \mathbb{F}_{q^n}^* \}$$

is $(r, i)$-regular fat exactly when this weight distribution is achieved (Smaldore et al., 6 Nov 2025).

Special cases:

• $(0, -)$-regular: scattered linear set (no points of weight $>1$)
• $(1, i)$-regular: $i$-club
• $(2, i)$-regular (rank $2i$): sets with complementary weights (Smaldore et al., 6 Nov 2025).
• For arbitrary $r > 0$, $r$-fat linearized polynomial requires exactly $r$ higher-weight points (weights possibly varying) (Bartoli et al., 2020).

2. Structural Properties and Generalizations

Regular fat linear sets encapsulate and generalize previously studied configurations:

• Scattered sets correspond to the case $r=0$.
• Clubs arise for $r=1$, arbitrary $i$, with numerous algebraic and combinatorial links.
• Sets with more than one higher-weight point (complementary-weight sets and general $r$-fat sets) extend the possible spectrum of configurations.

In $\mathrm{PG}(1,q^n)$, the weight spectrum controls both the size and algebraic structure of the underlying set. Classification by rank and field sizes yields that for small $n$ (notably $n\le 4$), the values of $r$ and $i$ are tightly constrained, and all equivalence classes are completely described (Bartoli et al., 2020).

Regularity imposes strict algebraic-design constraints. If all points have weight at least $2$, then, under further rank conditions, the linear set often admits extra field-linearity—a phenomenon captured by the notion of geometric field of linearity (Jena et al., 2021). This subtle distinction is essential for fine classification and for discriminating between genuinely new constructions and field-enlarged representations.

3. Explicit Constructions and Equivalence

3.1 Families from Subspace and Polynomial Data

A principal construction in $\mathrm{PG}(k-1, q^{2t})$ for $q$ odd, $t \geq 3$:

• $$E = \{w \in \mathbb{F}_{q^{2t}} : \operatorname{Tr}_{q^{2t}/q^t}(w) = 0\}$$, fix $w\in E^*$ with $\operatorname{N}_{q^t/q}(w^2) \neq (-1)^t$,
• $I \subseteq \mathbb{F}_{q^t}$ an $\mathbb{F}_q$-subspace of dimension $i$,
• Define $T = \{ x + w x^{q^s} : x \in I \}$ for $\gcd(s, t) = 1$,
• Let $U = T^k \subseteq (\mathbb{F}_{q^{2t}})^k$.

Then $\dim_q U = ki$ and $L_U$ is $((q^k-1)/(q-1), i)$-regular fat; the heavy points correspond exactly to the canonical subgeometry $\mathrm{PG}(k-1, q)$ (Smaldore et al., 6 Nov 2025).

3.2 Polynomial Representation

For $k=2$, explicit polynomials (via $q$-linearization) allow parameterization of all $(r, i)$-regular fat sets via kernel dimensions and root-counting in the associated Dickson matrices (Bartoli et al., 2020).

3.3 Equivalence Under Semilinear Group Action

Sets constructed via differing parameters $(s, w, I)$ are equivalent under $\Gamma\mathrm{L}(k, q^{2t})$ if and only if there exist an automorphism and a scalar satisfying explicit norm and scaling conditions involving $w, \tilde w$ and $I, \tilde I$. Inequivalence for differing $s$ or $t - s$ is detectable via invariants from the root structure (Smaldore et al., 6 Nov 2025).

4. Bounds, Instability, and Classification Results

• There exist no “exceptional” $r$-fat polynomials: for any fixed $r>0$, such families cannot persist across infinitely many extensions (Bartoli et al., 2020).
• Upper bounds for $r$ are explicit functions involving $q, n$, and the maximal fat point weight $W$: $r \le q^n - C_1 \sqrt{q^n} - C_2 q^{2W} + \cdots$, with constants depending on the context.
• For small parameters ($n \le 4$), the complete spectrum of admissible $(r, i)$ pairs is determined. For $n = 5$, explicit non-trace $1$-fat (club) examples are produced (Bartoli et al., 2020).

The strict instability of scatteredness is established: whenever a single point of weight $>1$ exists, the scattered property fails entirely, and the number of such points is necessarily bounded by field parameters.

5. Applications in Finite Geometry and Coding Theory

Regular fat linear sets play a pivotal role in constructing three-weight rank-metric codes. Specifically, an $(r,i)$-regular fat linear set of rank $\rho$ in $\mathrm{PG}(k-1, q^n)$, with $i < n$, determines:

• An F$_q$-subspace $U^{\perp'}$ of dimension $nk - \rho$,
• A rank-metric code $\mathcal{C} \subseteq \mathbb{F}_{q^n}^k$ with parameters $[nk-\rho,\, k,\, n-i]_{q^n/q}$,
• Codeword ranks: $n-i$ (occurring $r(q^n-1)$ times), $n-1$ (for remaining points), and $n$ (for the ambient space),
• Lower bounds for $r$ via MacWilliams identities, e.g. $$r\geq \frac{(q^{2\rho-nk}-1) \binom{n}{2}_q}{(q^n-1)\binom{i}{2}_q}$$ (Smaldore et al., 6 Nov 2025).

Such codes find applications in the construction and analysis of three-weight codes, MRD codes, and incidence combinatorics.

6. Interpolative Geometry and Algebraic Systems with Fat Points

The theory of regular fat linear sets is complemented by geometric interpolation problems with fat point schemes on projective surfaces. For instance, systems of divisors passing through collections of fat points on algebraic surfaces are governed by expected dimension counts and regularity phenomena:

• On surfaces degenerated to well-understood elliptic or ruled forms (e.g., the Atiyah surface), regularity of multiple fat points can be reduced to the one-point case,
• A conjectural equality between actual and expected dimensions for single fat points in characteristic zero determines regularity for all configurations (Zahariuc, 2017).
• In projective space, the Castelnuovo–Mumford regularity of schemes of fat points is sharply controlled by cluster configurations and the Segre–Trung–Catalisano–Valla bounds (Thien, 2016).

7. Open Problems and Future Directions

Several finely posed open questions remain:

• Existence of $(q^j+1, i)$-regular fat linear sets of rank $2i$ in $\mathrm{PG}(1, q^n)$ for $j>1$,
• Construction of $(r,i)$-regular fat linear sets in higher-dimensional settings whose heavy-point set deviates from the canonical subgeometry,
• Complete classification up to $\Gamma\mathrm{L}$ equivalence for fixed ranks and small parameters,
• Deeper links between field-linearity, point weights, and the interplay of algebraic and geometric fields of linearity (Jena et al., 2021),
• Applications in representability of sums of $q$-matroids and further families of few-weight rank-metric codes.

The development of the theory bridges finite geometry, algebraic geometry, combinatorics, and coding, providing new methods for constructing and classifying highly structured linear sets and the codes associated to them, and connects the arithmetic and geometric regularity properties of combinatorial point configurations (Smaldore et al., 6 Nov 2025, Bartoli et al., 2020, Zahariuc, 2017, Jena et al., 2021).