Range Avoidance for Low-Depth Circuits and Connections to Pseudorandomness

In the range avoidance problem, the input is a multi-output Boolean circuit with more outputs than inputs, and the goal is to find a string outside its range (which is guaranteed to exist). We show that well-known explicit construction questions such as finding binary linear codes achieving the Gilbert-Varshamov bound or list-decoding capacity, and constructing rigid matrices, reduce to the range avoidance problem of log-depth circuits, and by a further recent reduction [Ren, Santhanam, and Wang, FOCS 2022] to \(\mathsf {NC}^0_4\) circuits where each output depends on at most 4 input bits. On the algorithmic side, we show that range avoidance for \(\mathsf {NC}^0_2\) circuits can be solved in polynomial time. We identify a general condition relating to correlation with low-degree parities that implies that any almost pairwise independent set has some string that avoids the range of every circuit in the class. We apply this to \(\mathsf {NC}^0\) circuits, and to small-width CNF/DNF and general De Morgan formulae (via a connection to approximate-degree), yielding non-trivial small hitting sets for range avoidance in these cases.