Avoiding right angles and certain Hamming distances

TL;DR

This paper establishes new upper bounds on the size of subsets in finite fields avoiding right angles and certain Hamming distances, refuting previous conjectures and providing tight bounds for specific cases.

Contribution

It improves existing bounds on subset sizes avoiding right angles and triangles with right angles, and introduces bounds for subsets avoiding self-orthogonal differences and certain Hamming distances.

Findings

Largest subset avoiding right angles is at most O(n^{q-2})

Maximum size of subset avoiding right-angled triangles is O(n^{2q-2})

Bounds for subsets with non-self-orthogonal differences are asymptotically tight

Abstract

In this paper we show that the largest possible size of a subset of $\mathbb{F}_q^n$ avoiding right angles, that is, distinct vectors $x,y,z$ such that $x-z$ and $y-z$ are perpendicular to each other is at most $O(n^{q-2})$. This improves on the previously best known bound due to Naslund \cite{Naslund} and refutes a conjecture of Ge and Shangguan \cite{Ge}. A lower bound of $n^{q/3}$ is also presented. It is also shown that a subset of $\mathbb{F}_q^n$ avoiding triangles with all right angles can have size at most $O(n^{2q-2})$. Furthermore, asymptotically tight bounds are given for the largest possible size of a subset $A\subseteq \mathbb{F}_q^n$ for which $x-y$ is not self-orthogonal for any distinct $x,y\in A$. The exact answer is determined for $q=3$ and $n\equiv 2\pmod {3}$. Our methods can also be used to bound the maximum possible size of a binary code where no two codewords have Hamming distance divisible by a fixed prime $q$. Our lower- and upper bounds are asymptotically tight and both are sharp in infinitely many cases.