Biangular lines revisited

TL;DR

This paper investigates biangular line systems in Euclidean spaces, establishing lower bounds on their maximum size, exploring their connections to coding theory and association schemes, and extending the concept to multiangular cases.

Contribution

It provides new bounds for the maximum size of biangular lines and explores their relationships with binary codes, few-distance sets, and association schemes, including multiangular generalizations.

Findings

Maximum size of biangular lines is at least 2(d-1)(d-2).

The bound is sharp for dimensions 4, 5, and 6.

Connections to coding theory and association schemes are established.

Abstract

Line systems passing through the origin of the $d$ dimensional Euclidean space admitting exactly two distinct angles are called biangular. It is shown that the maximum cardinality of biangular lines is at least $2(d-1)(d-2)$, and this result is sharp for $d\in\{4,5,6\}$. Connections to binary codes, few-distance sets, and association schemes are explored, along with their multiangular generalization.