Investigation of binary sequences with the “trinomial property”

Summary form only given, as follows. It is well-known that if {a _k} is an m-sequence of period p=2ⁿ-1, then for every τ, 1⩽τ⩽p-1, {a_k}+{a_k+τ}={a_k+τ´} for some τ´, 1⩽τ´⩽p-1. We say that a binary sequence (b_k ) of period p has the trinomial property if there exists a pair (τ,τ´) with 1⩽τ<τ´⩽p-1 such that {b_k }+{b_k+τ}={b_k+τ´}. Many examples in addition to m-sequences exist. In this paper we derive constraints on sequences with the trinomial property, and present the results of comprehensive computer searches