Conflict-avoiding codes and near-primitive roots

An important role in the theory of conflict-avoiding codes is played by a quantity involving the multiplicative order of $4$ modulo $d$, with $d$ running over all positive divisors of an odd integer $n$. We establish results on its distribution as $n$ varies over the odd prime numbers, respectively odd integers, using techniques from the study of Artin's primitive root conjecture. We also point out some other interpretations of this quantity, with one of them making a connection with the factorization of $X^n-1$ into irreducibles over the finite field $\mathbb F_2$.