Code-based $[3,1]$-avoiders in finite affine spaces $\mathrm{AG}(n,2)$

TL;DR

This paper constructs large subsets of binary affine spaces that avoid certain small affine flats, using binary linear codes, to explore the minimal open case of avoiding $[3,1]$-flats.

Contribution

It provides explicit constructions of sets avoiding $[3,1]$-flats for many sizes, advancing understanding of affine space avoidance problems.

Findings

Constructed sets avoiding $[3,1]$-flats for exponentially many sizes.

Used binary linear codes with specific weight enumerators for constructions.

Contributed to the study of minimal avoidance configurations in affine spaces.

Abstract

The author, together with Nagy, studied the following problem on unavoidable intersections of given size in binary affine spaces. Given an $m$-element set $S\subseteq \mathbb{F}_2^n$, is there guaranteed to be a $[k,t]$-flat, that is, a $k$-dimensional affine subspace of $\mathbb{F}_2^n$ containing exactly $t$ points of $S$? Such problems can be viewed as generalizations of the cap set problem over the binary field. They conjectured that for every fixed pair $(k,t)$ with $k\ge 1$ and $0\le t\le 2^k$, the density of values $m\in \{0,...,2^n\}$ for which a $[k,t]$-flat is guaranteed tends to $1$. In this paper, motivated by the study of the smallest open case $(k,t)=(3,1)$, we present explicit constructions of sets in $\mathbb{F}_2^n$ avoiding $[k,1]$-flats for exponentially many sizes. These sets rely on carefully constructed binary linear codes, whose weight enumerators determine the size of the construction.