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On certain blocking sets and the minimum weight of the code of generalised polygons

Sebastian Petit, Geertrui Van de Voorde

TL;DR

The paper investigates the incidence-code of finite weak generalised $2m$-gons and establishes that the minimum weight equals $s+1$ under precise field-parameter conditions, with minimum-weight words characterized as distance traces $T_{d,x,y}$. The approach builds a bridge between blocking-set geometry and coding theory by introducing $\mathcal{X}$-blocking sets and weighted incidence vectors $\mathfrak{c}_v$ that encode distance-trace structure. The key contribution is a generalization of classical projective-plane results to not-necessarily-regular thick generalised polygons, including a detailed analysis of when distance traces yield minimal codewords. This advances understanding of how geometric configurations determine codeword supports and has potential implications for code design in combinatorial geometries and related incidence structures.

Abstract

In this paper, we study and characterise certain blocking sets in generalised polygons. This will allow us to derive new results about the minimum weight and minimum weight code words in the code generated by the rows of the incidence matrix of a generalised polygon over a field F.

On certain blocking sets and the minimum weight of the code of generalised polygons

TL;DR

The paper investigates the incidence-code of finite weak generalised -gons and establishes that the minimum weight equals under precise field-parameter conditions, with minimum-weight words characterized as distance traces . The approach builds a bridge between blocking-set geometry and coding theory by introducing -blocking sets and weighted incidence vectors that encode distance-trace structure. The key contribution is a generalization of classical projective-plane results to not-necessarily-regular thick generalised polygons, including a detailed analysis of when distance traces yield minimal codewords. This advances understanding of how geometric configurations determine codeword supports and has potential implications for code design in combinatorial geometries and related incidence structures.

Abstract

In this paper, we study and characterise certain blocking sets in generalised polygons. This will allow us to derive new results about the minimum weight and minimum weight code words in the code generated by the rows of the incidence matrix of a generalised polygon over a field F.

Paper Structure

This paper contains 16 sections, 29 theorems, 50 equations.

Table of Contents

  1. Introduction
  2. Motivation
  3. Background
  4. Weak generalised polygons
  5. Codes
  6. Thick generalised 3-gons and embeddable weak generalised polygons
  7. Blocking sets in projective space
  8. Blocking sets in weak generalised polygons
  9. Some elementary geometric properties
  10. Lines as blocking sets in weak generalised polygons
  11. Characterising X-blocking sets
  12. Regularity and distance traces
  13. Codes from weak generalised polygons
  14. A weighted incidence vector
  15. Proof of the main result
  16. ...and 1 more sections

Key Result

Corollary 1

(See e.g. Hendrik) Let $\Gamma$ be a point-line geometry with at least two points and the properties that: Then, $\Gamma$ is a weak generalised $n$-gon if and only if its incidence graph has diameter $n$ and girth $2n$.

Theorems & Definitions (74)

  • Definition 1
  • Definition 2
  • Remark 1
  • Remark 2
  • Remark 3
  • Corollary 1
  • Corollary 2
  • Theorem 1
  • Theorem 2
  • Definition 3
  • ...and 64 more