On the number of factorizations of polynomials over finite fields (Q2037165)

For a fixed prime power \(q\) and a positive integer \(m\) let denote by \(\Gamma_m\) the maximum number of distinct divisors of a polynomial over the finite field of~\(q\) elements of degree at most \(m\) and \(\Gamma_{n,n'}\) the maximum number of distinct factorizations into two polynomials of degree \(n\) and \(n'\) of any polynomial of degree at most \(n+n'\). The authors show \[\Gamma_m=2^{(m/\log_q m)(1+o(1))},\quad m\rightarrow \infty,\] and \[\Gamma_{n,n}=2^{(2n/\log_q n)(1+o(1))},\quad n\rightarrow \infty.\] These enumeration problems are motivated by applications in coding theory. Moreover, the authors characterize the maximal polynomials of degree at most \(m\) with \(\Gamma_m\) divisors and of degree at most \(2n\) with \(\Gamma_{n,n}\) factorizations into two polynomials of degree at most \(n\). Finally, they present expressions for the expected value and the variance of the number of factorizations.