Periodic Binary Sequences with the "Trinomial Property"

Abstract: Periodic binary sequences with the "trinomial property" are considered. Some necessary and sufficient conditions for "trinomial pairs" of a nonlinear sequence of period 2 n \Gamma 1 as well as classifications for trinomial pairs are derived. Complete searches have been done for 3 n 17. All trinomial pairs found on this range are listed. Index Terms: Periodic binary sequence, trinomial property, trinomial pair. 1 Introduction The m-sequences A = fa i g of degree n and period p = 2 n \Gamma 1 are characterized by the "cycle-andadd property": for each ø , 0 ø p there is a corresponding ø 0 , with fa i g + fa i+ø g = fa i+ø 0 g . Thus, the p cyclic shifts of fa i g together with the zero vector of length p form an n-dimensional subspace of F p 2 , the vector space of all p-component vectors over the two-element field F 2 . We say that a binary sequence A = fa i g of period p = 2 n \Gamma 1 has the trinomial property if there is (at least) one pair of positive i..