Honeycomb arrays

TL;DR

This paper investigates honeycomb arrays, a hexagonal grid analogue of Costas arrays, providing new theoretical bounds, summarizing computational searches, and presenting two new examples with 15 dots.

Contribution

It introduces a theorem that significantly restricts honeycomb array possibilities and reports new examples, advancing understanding of their structure and limitations.

Findings

Number of dots in a honeycomb array must be odd

Two new honeycomb arrays with 15 dots found

Theoretical bounds on honeycomb arrays improved

Abstract

A honeycomb array is an analogue of a Costas array in the hexagonal grid; they were first studied by Golomb and Taylor in 1984. A recent result of Blackburn, Etzion, Martin and Paterson has shown that (in contrast to the situation for Costas arrays) there are only finitely many examples of honeycomb arrays, though their bound on the maximal size of a honeycomb array is too large to permit an exhaustive search over all possibilities. The present paper contains a theorem that significantly limits the number of possibilities for a honeycomb array (in particular, the theorem implies that the number of dots in a honeycomb array must be odd). Computer searches for honeycomb arrays are summarised, and two new examples of honeycomb arrays with 15 dots are given.