Blocking Sets in Generalised Polygons

• Blocking sets in generalised polygons are point sets intersecting every line or substructure, bridging finite geometry with coding theory.
• The study employs group-theoretic methods and explicit coordinatizations of triad and triangle configurations to classify index-3 blocking sets.
• Applications include constructing incidence codes and identifying minimum-weight codewords, with implications for error-correcting codes and projective designs.

Blocking sets in generalised polygons constitute a central concept in the intersection of finite geometry, combinatorial design, and coding theory. In the context of a finite incidence geometry—such as projective planes and higher-rank generalised polygons—a blocking set is a set of points intersecting each line or, more generally, each of a specified collection of substructures. The paper of minimal blocking sets, their classification by index (the minimal number of lines covering them), and their explicit characterisation in both projective planes and higher generalised polygons connects deep combinatorial properties with applications to the theory of linear codes generated by incidence matrices.

1. Preliminaries: Generalised Polygons, Blocking Sets, and Index

Let $\Gamma=(P,L,I)$ be a finite (weak or thick) generalised $n$–gon of order $(s,t)$, where each line meets $s+1$ points and each point lies on $t+1$ lines. The case $n=3$ recovers the projective plane $PG(2,q)$. The incidence graph of $\Gamma$ is bipartite, with girth $2n$ and diameter $n$ when $\Gamma$ is thick.

A set $C\subseteq P$ is called a blocking set if it intersects every member of a chosen family of point-sets—typically, all lines (projective planes) or other natural geometric substructures (generalised polygons). In projective geometry, a proper blocking set is one that contains no line as a subset. The index of a blocking set $S$ is the minimal number of lines whose union covers $S$. For projective planes, $\mathrm{index}(S)\ge3$, with particular interest in the structure of index-3 blocking sets.

2. Explicit Classification in Projective Planes: The Case of Index-3

In $PG(2,q)$, a proper blocking set $S$ is minimal if no proper subset is also a blocking set. Cherowitzo and Holder established the following complete characterisation for blocking sets of index 3 (Cherowitzo et al., 2012):

• $S$ is covered by exactly three lines, which are either concurrent (a triad) or form a triangle.
• The possible sizes of $S$ are:
  1. $2q$
  2. $3(q-1)$
  3. $3q+1-m$, where $(GF(q),+)\to m$ and $q>2$
  4. $3q - m$, where $(GF(q)^*,\cdot)\to m$; here, the group-theoretic notation means there exist non-empty subsets $A,B,C$ with $0\notin A+B+C$, maximality, and $|A|+|B|+|C|=m$.

Cases (iii) and (iv) are instances of Rédei blocking sets, where the blocking set admits a "Rédei line" intersected by many points of $S$. The possible values for the size of the Rédei line are either $q$ or $q-1$ points of $S$.

The triad and triangle configurations permit explicit coordinatisations and correspond to concrete group-theoretic problems involving additive/multiplicative sum-free sets and coset partitions. For example, in the triad case, the set $S$ is distributed over three concurrent lines; in the triangle case, $S$ covers all but a specific coset arrangement within the triangle formed by three lines.

3. Constructions in Generalised Polygons and Minimal Blocking Sets

In a thick weak generalised $2m$–gon of order $(s,t)$, the notion of a blocking set is naturally extended. Let

$\mathcal X = \{\,P_{\le2m-2}(v)\mid v \in P\,\}$

where $P_{\le2m-2}(v)$ denotes the set of all points at distance at most $2m-2$ from $v$. An $\mathcal X$–blocking set $C$ is a point set satisfying $C \cap P_{\le2m-2}(v) \neq \emptyset$ for all $v\in P$.

Every line is an $\mathcal X$–blocking set. The minimal size of such blocking sets is $s+1$. The minimal blocking sets are precisely the distance-$d$ traces $T_{d,x,y} = P_d(x) \cap P_{2m-d}(y)$ for $1\le d\le m$ and $x,y$ opposite points or lines (i.e., $\delta(x,y)=2m$).

If $s<t$ (for example, in certain quadrangles), then all minimal blocking sets correspond to $d=1$, and thus the only minimal $\mathcal X$–blocking sets are the sets of points of a line. These results extend classical characterisations in projective planes to the context of higher generalised polygons (Petit et al., 10 Nov 2025).

4. Blocking Sets and the Coding Theory Connection

Let $C(\Gamma)$ be the code over a field $\mathbb F$ generated by the incidence matrix of the generalised polygon $\Gamma$: each codeword corresponds to a formal sum of indicator vectors of lines. The minimum weight $d_{\min}(C(\Gamma))$ is the minimal size of the support of a nonzero codeword.

It is proven that for a thick generalised $2m$–gon of order $(s,t)$ (with $s\le t$ and $m>1$) and over any $\mathbb F$ where certain sums $\sum_{j=0}^k(-s)^j$ do not vanish, one has

$d_{\min}(C(\Gamma)) = s+1$

and every minimum-weight codeword corresponds (up to scalar) to the indicator of the point set of a line, or more generally, of a minimal distance-$d$ trace $T_{d,x,y}$. In projective planes, this recovers the classical fact; in split Cayley hexagons, further families of minimum-weight codewords arise from nondegenerate conics corresponding to distance-2 or distance-3 traces. These results reflect a precise geometric–algebraic link between minimal blocking sets and minimals of the incidence code (Petit et al., 10 Nov 2025).

5. Concrete Classifications and Examples: PG(2,7) and Beyond

The classification in $PG(2,7)$ is complete:

• Size 12: exactly two minimal index-3 blocking sets—one projective triangle Rédei set ($d=3$), and one non-Rédei set covered by three concurrent lines.

• Size 13: a unique (up to collineation) Rédei index-3 blocking set, where the Rédei line contains $q-1=6$ points.
• Size 14: eight projectively inequivalent Rédei index-3 sets (triad and triangle types for various values of $|A|$ or $|B|$), and three non-Rédei index-3 sets classified by Innamorati–Maturo.

In higher-rank generalised polygons, explicit constructions involve subgroup–coset decompositions: for example, for $d\mid(q-1)$, one can construct all Rédei index-3 sets of size $2q+1-d$ by taking two distinct cosets of a subgroup $H$ of order $d$ in $GF(q)^*$ and extending to a blocking set via points on lines at infinity (Cherowitzo et al., 2012).

6. Methodological Insights and Group-Theoretic Reformulations

The group-theoretic perspective, formalised by Cameron, plays a crucial role: blocking sets of a given index correspond to multi-sum–free (or product–free) subsets in abelian (additive or multiplicative) groups. For index-3 sets, these are characterisations of $(G,\to m)$ for $G=GF(q)$ or $GF(q)^*$. This perspective, complemented by Kneser’s theorem for group subsets, supports both the classification for small index and the identification of subgroup–coset constructions in higher index scenarios.

Extensions to larger $q$ and higher-dimensional geometries remain open, especially the classification of minimal blocking sets of index 4 in $PG(2,q)$ and analogous uniqueness results. In the context of thick generalised polygons, subgroup–coset decompositions of Moufang-type groups are conjectured to yield further families of small-index blocking sets.

7. Open Problems and Directions

Key unresolved questions include:

• Full classification of minimal blocking sets of index four in projective planes.
• Determining the uniqueness (up to collineation) of blocking sets of a given index for larger field sizes.
• Extension of the theory to blocking sets in generalised polygons of higher rank, higher-dimensional projective spaces, or in specific polar spaces (symplectic, orthogonal).
• Elucidating the precise geometric conditions under which non-line minimal-weight codewords arise in the incidence codes of generalised polygons of exceptional type.

The development of theory connecting blocking sets, the combinatorics of sum/product sets in finite groups, and the algebraic geometry of incidence structures continues to motivate new research at the interface of finite geometry and coding theory.

References:

Cherowitzo–Holder, "Blocking Sets of Index Three" (Cherowitzo et al., 2012); "On certain blocking sets and the minimum weight of the code of generalised polygons" (Petit et al., 10 Nov 2025).