Counting perfect polynomials

Let A∈F2[T]. We say A is perfect if A coincides with the sum of all of its divisors in F2[T]. We prove that the number of perfect polynomials A with |A|≤x is Oϵ(x1/12+ϵ) for all ϵ>0, where |A|=2deg⁡A. We also prove that every perfect polynomial A with 1<|A|≤1.6×1060 is divisible by T or T+1; that is, there are no small “odd” perfect polynomials.