Periodic binary sequences with the "trinomial property"

The m-sequences A={a_i} of degree n and period p=2ⁿ-1 are characterized by the "cycle-and-add property": for each τ, 0<τi}+{a_i+τ}={a_i+τ\´}. Thus, the p cyclic shift of {a_i} together with the zero vector of length p form an n-dimensional subspace of F₂^p, the vector space of all p-component vectors over the two-element field F₂. We say that a binary sequence A={a_i} of period p=2^n-1 has the trinomial property if there is (at least) one pair of positive integers, τ and τ\´, such that {a_i}+{a_i+τ}={a_i+τ\´}, where 0<τ<τ\´τ\´+x^τ+1). The pair (r, r\´) is called a trinomial pair of A={a_i}. Two trinomial pairs (τ₁, τ₁\´) and (τ₂, τ₂\´) of A are called equivalent if τ₂≡2^jτ₁ (mod p) for some non-negative integer j; otherwise they are called distinct. We list some necessary and sufficient conditions for determining the trinomial pairs of a nonlinear sequence of period p.