Honeycomb arrays (Q612962)

Summary: A honeycomb array is an analogue of a Costas array in the hexagonal grid; they were first studied by {|it S.W. Golomb} and \textit{H. Taylor} in [``Construction and properties of Costas arrays,'' Proc. IEEE 72, No.\,9, 1143--1163 (1984; Zbl 1200.05043)]. A recent result of \textit{S.R. Blackburn}, \textit{T. Etzion}, \textit{K.M. Martin} and \textit{M.B. Paterson} [``Two-dimensional patterns with distinct differences --- Constructions, bounds and maximal anticodes,'' IEEE Trans. Inf. Theory 56, 1216--1229 (2010)] 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.