Generalised 4-Corner Set

• Generalised 4-Corner Sets are point sets in finite projective space that span the ambient space and ensure no four points lie in a plane.
• They serve as higher-order analogues of caps and are closely related to saturating sets, covering codes, and Sidon sets in binary fields.
• Research explores extremal parameters and construction methods, with implications in combinatorial geometry and linear coding theory.

Searching arXiv for recent and directly relevant papers on generalized 4-corner / 4-general sets. A generalised 4-corner set, more commonly termed a 4-general set, is a point set $X \subseteq {\rm PG}(n,q)$ in finite projective space that spans the ambient space and satisfies the incidence prohibition that no four points of $X$ are coplanar. In the formulation studied by Pavese, the set is required to generate the whole ${\rm PG}(n,q)$, and completeness means maximality under inclusion among such point sets (Pavese, 2023). The subject lies at the intersection of finite projective geometry, extremal combinatorics, and coding theory, where 4-general sets provide a higher-order analogue of caps and are closely related to saturating sets, covering codes, and, over binary fields, Sidon sets.

1. Definition and basic geometric meaning

Let $X \subseteq {\rm PG}(n,q)$ be a point-set. A 4-general set is defined by two conditions: $(1)$ $\langle X\rangle = {\rm PG}(n,q)$; $(2)$ no four points of $X$ lie in the same projective plane (Pavese, 2023).

Equivalently, for every choice of four distinct points $P_1,P_2,P_3,P_4 \in X$, the span is not a plane. In the terminology used in the source, a 4-general set must span the whole ambient projective space and exclude every coplanar 4-tuple. This strengthens the usual cap condition. In particular, any three points of $X$ are in general position, and if $X$0, then $X$1 is a cap, since the presence of three collinear points would force any fourth point to lie in a plane with them, contradicting the 4-general condition (Pavese, 2023).

A 4-general set is called complete if it is maximal with respect to inclusion: there is no larger 4-general set $X$2 with $X$3 (Pavese, 2023). Pavese denotes by $X$4 the maximum size of a complete 4-general set in $X$5, and by $X$6 the minimum size of a complete 4-general set (Pavese, 2023).

The geometric constraint “no four on a plane” is the defining structural feature. A plausible implication is that the topic may be viewed as a projective-space analogue of forbidding affine parallelogram-type configurations, but the formal theory developed in the cited work is entirely projective and incidence-theoretic.

2. Completeness and extremal parameters

The principal extremal invariants are the largest and smallest sizes of complete 4-general sets. For an arbitrary 4-general set $X$7, Pavese derives a quadratic inequality: $X$8 From this one obtains the upper bound

$X$9

Equality occurs only in the two classical cases ${\rm PG}(n,q)$0, when ${\rm PG}(n,q)$1 is an elliptic quadric in ${\rm PG}(n,q)$2, and ${\rm PG}(n,q)$3, when ${\rm PG}(n,q)$4 is the unique 11-cap in ${\rm PG}(n,q)$5 (Pavese, 2023).

Completeness imposes a complementary covering constraint. Every point of ${\rm PG}(n,q)$6 must lie on at least one plane spanned by three points of ${\rm PG}(n,q)$7. Using this fact, Pavese proves that a complete 4-general set satisfies

${\rm PG}(n,q)$8

where ${\rm PG}(n,q)$9 is the unique positive real root of

$X \subseteq {\rm PG}(n,q)$0

Equivalently,

$X \subseteq {\rm PG}(n,q)$1

In particular,

$X \subseteq {\rm PG}(n,q)$2

as $X \subseteq {\rm PG}(n,q)$3 or $X \subseteq {\rm PG}(n,q)$4 grows (Pavese, 2023).

These bounds place the subject within extremal finite geometry. The upper estimate constrains how large a plane-avoiding spanning set can be, while the lower estimate shows that maximality forces substantial size because every external point must be “covered” by a plane through three internal points.

3. Infinite constructions and near-extremal families

Pavese constructs several infinite families of complete 4-general sets whose cardinalities are close to the theoretical upper bound in specific regimes (Pavese, 2023).

For $X \subseteq {\rm PG}(n,q)$5, in $X \subseteq {\rm PG}(n,q)$6 and for every $X \subseteq {\rm PG}(n,q)$7, there exists a complete 4-general set of size $X \subseteq {\rm PG}(n,q)$8. The construction uses the cyclic model of projective space, writing

$X \subseteq {\rm PG}(n,q)$9

with $(1)$0 a primitive element of $(1)$1, and taking the orbit of a subgroup of order $(1)$2 under the descriptive projectivity $(1)$3. The proof that no four points are coplanar is combined with completeness via Hermitian-curve intersection arguments (Pavese, 2023). This yields

$(1)$4

The case $(1)$5 recovers the classical 11-cap in $(1)$6, and $(1)$7 gives a complete 4-general set of size $(1)$8 in $(1)$9 (Pavese, 2023).

For $\langle X\rangle = {\rm PG}(n,q)$0, in $\langle X\rangle = {\rm PG}(n,q)$1 and for each $\langle X\rangle = {\rm PG}(n,q)$2, there exists a complete 4-general set of size $\langle X\rangle = {\rm PG}(n,q)$3. The construction again uses the cyclic model and three Hermitian surfaces to establish completeness (Pavese, 2023). Consequently,

$\langle X\rangle = {\rm PG}(n,q)$4

The following table summarizes the main infinite families explicitly described.

Ambient space	Field condition	Size of complete 4-general set
$\langle X\rangle = {\rm PG}(n,q)$5	$\langle X\rangle = {\rm PG}(n,q)$6	$\langle X\rangle = {\rm PG}(n,q)$7
$\langle X\rangle = {\rm PG}(n,q)$8	$\langle X\rangle = {\rm PG}(n,q)$9	$(2)$0
$(2)$1	$(2)$2	$(2)$3

These families show that the extremal theory is not purely asymptotic: highly structured algebraic constructions exist in several parameter ranges, often tied to cyclic models, Hermitian varieties, or explicit group actions.

4. Small fields, classification results, and binary behavior

The binary case has a particularly transparent reformulation. In $(2)$4, 4-general sets correspond to Sidon sets in $(2)$5 (Pavese, 2023). This places the topic within additive combinatorics as well as projective geometry. The correspondence is emphasized in Pavese’s discussion of binary spaces and small dimensions, where complete sets are classified for $(2)$6 (Pavese, 2023).

Examples explicitly listed in the source include the following. In $(2)$7, the unique 5-cap is a frame. In $(2)$8, the 7-cap is a twisted cubic. Up to $(2)$9, one encounters a 13-cap arising from a frame plus one extra point (Pavese, 2023). These examples illustrate the strong rigidity of low-dimensional binary instances.

For $X$0 with $X$1, the complete 4-general sets are exactly the $X$2-arcs, such as conics or twisted cubics, together with two sporadic cases, including the 11-cap for $X$3 and the 21-cap for $X$4 (Pavese, 2023). The classification indicates that in small dimensions the 4-general condition often singles out classical algebraic curves and a limited number of exceptional configurations.

This binary and small-field behavior is significant because it connects 4-general sets with well-studied discrete structures. Over $X$5, the incidence condition becomes equivalent to a Sidon-type additive constraint, while in low-dimensional projective spaces the resulting configurations admit nearly complete classification.

5. Transitive constructions and group actions

A notable family in $X$6, for $X$7, is obtained through a transitive construction of size $X$8 (Pavese, 2023). The ambient vector space is taken as

$X$9

with coordinate structure

$P_1,P_2,P_3,P_4 \in X$0

A subgroup $P_1,P_2,P_3,P_4 \in X$1 is generated by the three coordinate scalings

$P_1,P_2,P_3,P_4 \in X$2

where $P_1,P_2,P_3,P_4 \in X$3 and $P_1,P_2,P_3,P_4 \in X$4 satisfies $P_1,P_2,P_3,P_4 \in X$5 in $P_1,P_2,P_3,P_4 \in X$6. The resulting group has order $P_1,P_2,P_3,P_4 \in X$7 and acts semiregularly on three twisted cubics $P_1,P_2,P_3,P_4 \in X$8, each lying in a distinct 3-space. Their union

$P_1,P_2,P_3,P_4 \in X$9

is a single $X$0-orbit of size $X$1 (Pavese, 2023).

The geometric verification has two parts. First, no four points of $X$2 lie in a plane; second, completeness is established either through covering-radius arguments or by checking that every point of $X$3 lies on a plane determined by three points of $X$4 (Pavese, 2023). The source highlights that the three containing 3-spaces meet pairwise in the same line, which is an “imaginary chord” to each twisted cubic.

This construction is important because it combines symmetry, transitivity, and completeness in a nontrivial way. A plausible implication is that group-theoretic methods may be especially effective when one seeks large, structured 4-general sets with controllable incidence geometry.

6. Relations to coding theory, saturation, and open directions

Pavese places 4-general sets in a broader framework of coding-theoretic and geometric structures (Pavese, 2023). Any 4-general set $X$5 with $X$6 corresponds, via a parity-check matrix whose columns are point representatives, to a nonextendable linear code $X$7 of covering radius $X$8 and minimum distance $X$9 (Pavese, 2023). Two classical perfect-code exceptions are explicitly identified: $X$00, corresponding to the elliptic quadric in $X$01, and $X$02, corresponding to the Mathieu-23 cap in $X$03 (Pavese, 2023).

The same source notes that 4-general sets are examples of 2-saturating sets: every point of $X$04 lies in a plane spanned by three points of $X$05 (Pavese, 2023). This is exactly the covering property that underlies the lower-bound argument for complete sets. Over $X$06, the affine analogue in $X$07 is described by Sidon sets in $X$08, linking the theory to the literature on maximal Sidon sets (Pavese, 2023).

Further geometric constructions can be obtained by puncturing. For example, intersecting the Desarguesian ovoid in $X$09 with a hyperplane yields complete 4-general sets in $X$10 of size $X$11 (Pavese, 2023). This indicates that 4-general sets can arise from classical varieties through dimension-reduction procedures.

Several open problems are identified in the source. These include the exact determination of $X$12 for general $X$13, the existence of complete 4-general sets of $X$14 points in $X$15 for fixed $X$16, and the classification of complete 4-general sets meeting at most 4 points on any 3-space (Pavese, 2023). These questions reflect the present status of the area: the foundational extremal theory is established, several rich infinite families are known, but the global classification and sharp asymptotics remain incomplete.

In this sense, the generalised 4-corner set is best understood as a finite-geometric incidence object defined by a fourth-order noncoplanarity condition, whose significance derives from its dual role as an extremal configuration in projective space and as a structural model for nonextendable covering codes and saturating sets (Pavese, 2023).