  Range Avoidance (AVOID) is a total search problem where, given a Boolean
circuit $C\colon\{0,1\}^n\to\{0,1\}^m$, $m>n$, the task is to find a
$y\in\{0,1\}^m$ outside the range of $C$. For an integer $k\geq 2$,
$\mathrm{NC}^0_k$-AVOID is a special case of AVOID where each output bit of $C$
depends on at most $k$ input bits. While there is a very natural randomized
algorithm for AVOID, a deterministic algorithm for the problem would have many
interesting consequences. Ren, Santhanam, and Wang (FOCS 2022) and Guruswami,
Lyu, and Wang (RANDOM 2022) proved that explicit constructions of functions of
high formula complexity, rigid matrices, and optimal linear codes, reduce to
$\mathrm{NC}^0_4$-AVOID, thus establishing conditional hardness of the
$\mathrm{NC}^0_4$-AVOID problem. On the other hand, $\mathrm{NC}^0_2$-AVOID
admits polynomial-time algorithms, leaving the question about the complexity of
$\mathrm{NC}^0_3$-AVOID open.
  We give the first reduction of an explicit construction question to
$\mathrm{NC}^0_3$-AVOID. Specifically, we prove that a polynomial-time
algorithm (with an $\mathrm{NP}$ oracle) for $\mathrm{NC}^0_3$-AVOID for the
case of $m=n+n^{2/3}$ would imply an explicit construction of a rigid matrix,
and, thus, a super-linear lower bound on the size of log-depth circuits.
  We also give deterministic polynomial-time algorithms for all
$\mathrm{NC}^0_k$-AVOID problems for $m\geq n^{k-1}/\log(n)$. Prior work
required an $\mathrm{NP}$ oracle, and required larger stretch, $m \geq
n^{k-1}$.
