  We construct efficient implicit-explicit BDF$k$ scalar auxiliary variable
(SAV) schemes for general dissipative systems. We show that these schemes are
unconditionally stable, and lead to a uniform bound of the numerical solution
in the norm based on the principal linear operator in the energy. Based on this
uniform bound, we carry out a rigorous error analysis for the $k$th-order
$(k=1,2,3,4,5)$ SAV schemes in a unified form for a class of typical Allen-Cahn
type and Cahn-Hilliard type equations. We also present numerical results
confirming our theoretical convergence rates.
