Costas arrays, Golomb rulers and wavelength isolation sequence pairs

This thesis studies two combinatorial objects arising from applications in digital information processing. We firstly consider “wavelength isolation sequence pairs” (WISPs), a type of binary sequence pair introduced by Golay in 1951 but largely neglected since. Two previously overlooked examples of such sequence pairs are presented. We construct all known examples of WISPs from perfect Golomb rulers, and give partial classification results. We secondly consider Costas arrays, a generalisation of Golomb rulers dating from 1965. We examine whether a Costas array can contain every possible toroidal distance vector; contrary to claims elsewhere, this is still an open question. We constrain the (non-toroidal) distance vectors in Costas arrays by introducing “mirror pairs”. Structural properties of all Costas arrays are established via the number and type of their mirror pairs, with stronger results for G-symmetric Costas arrays, Welch Costas arrays and Golomb Costas arrays.

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