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

TL;DR

This paper characterizes the smallest even sets in projective spaces as cylinders with hyperoval bases and advances the understanding of dual projective geometric codes, specifically focusing on minimum weight codewords and their structure.

Contribution

It proves the conjecture for even q, relates minimum codeword weight to the 2-dimensional case, and shows that verifying the conjecture for n=3 implies it for all n.

Findings

Smallest even sets are cylinders with hyperoval bases.

Minimum weight of codes scales with q^{n-2} times the 2D case.

Verification for n=3 implies the conjecture for all n.

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.