In this paper, we present a construction for binary sequences {s(t)} of period N= ulcorner-1 for an odd prime ulcornerbased on the polynomial (z+1)d+az d +b, and discuss them in some cases of parameters p, m, d, a and b. We show that new sequences from our construction are balanced or almost balanced, and have optimal three-level autocorrelation. We also derive the distribution of autocorrelation values they take on. The sequences satisfy constant-on-the-coset property, and we will show that there are more than one characteristic phases with constant-on-the-coset property. Some other interesting properties of these sequences will be presented. Results of an extensive computer search are summarized.