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Even Sets and Dual Projective Geometric Codes: A Tale of Cylinders

Sam Adriaensen

TL;DR

This paper characterizes the smallest even sets in projective space $\\mathrm{PG}(n,q)$ as cylinders with a hyperoval base when $q$ is even, and ties this to the minimum weight of the dual projective geometric codes $\\mathcal{C}_1(n,q)^\\perp$. It establishes that the minimum weight satisfies $d(\\mathcal{C}_1(n,q)^\\perp)=q^{n-2}d(\\mathcal{C}_1(2,q)^\\perp)$ for $n>2$, and proves that cylinder codewords achieve this bound; it further proves a reduction to the three-dimensional case and shows that, for even $q$, minimum-weight codewords are cylinders. Finally, it relates the problem to the classification of smallest even sets via hyperovals, and provides a framework connecting 2D and higher-dimensional dual codes through a variance-based, incidence-geometric approach. These results advance understanding of dual projective geometric codes and offer a path to resolving minimum-weight questions by reducing to the 2D case and leveraging hyperoval classifications.

Abstract

In this paper, we prove that the smallest even sets in ${\rm PG}(n,q)$, i.e. sets that intersect every line in an even number of points, are cylinders with a hyperoval as base. This fits into a more general study of dual projective geometric codes. Let $q$ be a prime power, and define $\mathcal C_k(n,q)^\perp$ as the kernel of the $k$-space vs. point incidence matrix of ${\rm PG}(n,q)$, seen as a matrix over the prime order subfield of $\mathbb F_q$. Determining the minimum weight of this linear code is still an open problem in general, but has been reduced to the case $k=1$. There is a known construction that constructs small weight codewords of $\mathcal C_1(n,q)^\perp$ from minimum weight codewords of $\mathcal C_1(2,q)^\perp$. We call such codewords cylinder codewords. We pose the conjecture that all minimum weight codewords of $\mathcal C_1(n,q)^\perp$ are cylinder codewords. This conjecture is known to be true if $q$ is prime. We take three steps towards proving that the conjecture is true in general: (1) We prove that the conjecture is true if $q$ is even. This is equivalent to our classification of the smallest even sets. (2) We prove that the minimum weight of $\mathcal C_1(n,q)^\perp$ is $q^{n-2}$ times the minimum weight of $\mathcal C_1(2,q)^\perp$, which matches the weight of cylinder codewords. Thus, we completely reduce the problem of determining the minimum weight of $\mathcal C_1(n,q)^\perp$ to the case $n=2$. (3) We prove that if the conjecture is true for $n=3$, it is true in general.

Even Sets and Dual Projective Geometric Codes: A Tale of Cylinders

TL;DR

This paper characterizes the smallest even sets in projective space as cylinders with a hyperoval base when is even, and ties this to the minimum weight of the dual projective geometric codes . It establishes that the minimum weight satisfies for , and proves that cylinder codewords achieve this bound; it further proves a reduction to the three-dimensional case and shows that, for even , minimum-weight codewords are cylinders. Finally, it relates the problem to the classification of smallest even sets via hyperovals, and provides a framework connecting 2D and higher-dimensional dual codes through a variance-based, incidence-geometric approach. These results advance understanding of dual projective geometric codes and offer a path to resolving minimum-weight questions by reducing to the 2D case and leveraging hyperoval classifications.

Abstract

In this paper, we prove that the smallest even sets in , i.e. sets that intersect every line in an even number of points, are cylinders with a hyperoval as base. This fits into a more general study of dual projective geometric codes. Let be a prime power, and define as the kernel of the -space vs. point incidence matrix of , seen as a matrix over the prime order subfield of . Determining the minimum weight of this linear code is still an open problem in general, but has been reduced to the case . There is a known construction that constructs small weight codewords of from minimum weight codewords of . We call such codewords cylinder codewords. We pose the conjecture that all minimum weight codewords of are cylinder codewords. This conjecture is known to be true if is prime. We take three steps towards proving that the conjecture is true in general: (1) We prove that the conjecture is true if is even. This is equivalent to our classification of the smallest even sets. (2) We prove that the minimum weight of is times the minimum weight of , which matches the weight of cylinder codewords. Thus, we completely reduce the problem of determining the minimum weight of to the case . (3) We prove that if the conjecture is true for , it is true in general.
Paper Structure (12 sections, 24 theorems, 38 equations)

This paper contains 12 sections, 24 theorems, 38 equations.

Table of Contents

  1. Introduction
  2. Projective geometric codes
  3. Bounds on the minimum weight of the dual code
  4. The cylinder construction and a conjecture
  5. New results
  6. Cylinder codewords have minimum weight
  7. Reduction to three dimensions
  8. Even prime powers and even sets
  9. Organisation of the paper and proof techniques
  10. Proof of Theorem
  11. Proof of Theorem
  12. Proof of Theorem

Key Result

Theorem 1.1

If $k > 1$, then Moreover, the minimum weight codewords of $\mathcal{C}_k(n,q)^\perp$ arise exactly as follows: Take an $(n-k+1)$-space $\pi$, a codeword $c$ in the dual code of points and lines in $\pi$, and extend $c$ to a codeword of $\mathcal{C}_k(n,q)^\perp$ by putting $c(P) = 0$ for all points $P \notin \pi$.

Theorems & Definitions (40)

  • Theorem 1.1: lavrauwstormevandevoorde2008kspaces
  • Theorem 1.2: bagchiinamdar lavrauwstormevandevoorde2008kspaces
  • Theorem 1.3: calkinkeyderesmini
  • Theorem 1.4: bagchiinamdar
  • Theorem 1.5: csajbok2025lower
  • Theorem 1.6: LavrauwStormeVandeVoorde2010
  • Theorem 1.7
  • Definition 1.8
  • Proposition 1.9: bagchiinamdar
  • Conjecture 1.10
  • ...and 30 more