A Shifting Property of Some Costas Sequences

There is a conjecture of Golomb and Taylor that asserts that the Welch construction for Costas sequences with length \(p-1\), \(p\) prime, is the only one with the property of single periodicity.

In the present paper we present and prove a weaker conjecture: the Welch construction is the only one with the property that its differences are a shift of the original sequence.