Expected 2-adic complexity of finite random binary sequences

Determine whether the expected value of the Nth 2-adic complexity over all binary sequences of length N satisfies the conjectured asymptotic relation analogous to the expected linear-complexity formula, namely E2-adic_N = N/2 + O(1).

Background

For a binary sequence of length N, the Nth 2-adic complexity is the binary logarithm of the least rational complexity of a rational representation of the sequence. The paper compares its average behavior with the classical result for Nth linear complexity, whose expected value is N/2 + O(1).

Prior experimental results suggested that the expected Nth 2-adic complexity is close to N/2, but the authors explain that the corresponding theoretical result remained unresolved. The paper proves the weaker but substantial estimate E2-adic_N = N/2 + O(log N), so the sharper bounded-deviation form remains the explicitly identified open problem.