ℋ∞norm bounds are useful in control design for disturbance rejection. For discrete systems, ℋ∞norm optimizing controls can be characterized via the solution of the Discrete Algebraic Riccati Equation, (DARE). In addition, the DARE can be used to characterize controls that guarantee a uniform ℋ∞norm bound. In this paper we identify an entire class of state feedback controls that guarantee an ℋ∞norm bound, and provide new lower bounds for the ℋ∞norm of stable systems, as well as for the best achievable ℋ∞norm using state feedback controls.