Andreev et al. (3) gave constructions of Boolean functions (computable by polynomial-size circuits) with large lower bounds for read-once branching rpogram (1-b.p.'s): a function in P with the lower bound 2~n-polylog(n), a function in quasipolynomial time with the lower bound 2~n-O(log n), and a function in LINSPACE with the lower bound 2~n-log n-O(1). We point out alternative, much simpler constructions of such Boolean functions by applying the idea of almost k-wise independence more directly, without the use of discrepancy set generators for large affine subspaces; our consrrucitons are obtianed by derandomizing the probabilistic proofs of existence of the corresponding combinatorial objects. The simplicity of our new constructions also allows us to observe that there exists a Boolean function in AC deg (2) (computable by a depth 3, polynomial-size circuit over the basis (direct +, 1)) with the optimal lower bound 2~n-log n-O(1) for 1-b.p.'s.
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