On an Erd{o}s-type conjecture on F₂[x]

In 1970, R. B. Crittenden and C. L. Vanden Eynden{CV1970} proved that any system of n congruence classes covering positive integers 1,2,... 2ⁿ covers all integers, which was originally conjectured by P. Erd{o}s{Erdos1962} in 1962, who first proved a weaker version of this conjecture, replacing 2ⁿ with $C(n)$, where $C(n)$ is a constant determined by n. The constant 2ⁿ here cannot be improved, since one can find n congruences covering any given 2ⁿ consecutive integers with any one given exception among them exactly. In 2019, P. Balister, B. Bollob\'as, R. Morris, J. Sahasrabudhe and M. Tiba{BBST2020} gave a simpler proof of this conjecture also by the proof by contradiction,including constructing polymials using parameters from the counter-example congruences.