Expected 2-adic complexity within a bounded deviation of half the sequence length

Establish that the expected value of the Nth 2-adic complexity over all binary sequences of length N satisfies the analogue of the linear-complexity estimate E_lin_N = N/2 + O(1), namely E_2-adic_N = N/2 + O(1), rather than only the currently established bound E_2-adic_N = N/2 + O(log N).

Background

The paper studies the average and asymptotic behavior of the 2-adic complexity of finite and infinite binary sequences. For linear complexity, the expected value over binary sequences of length N is known to be N/2 + O(1). Experimental evidence suggests that the expected Nth 2-adic complexity obeys an analogous estimate, but an exact counting formula comparable to the one available for linear complexity is not known.

The authors prove the weaker estimate E_2-adic_N = N/2 + O(log N), substantially improving earlier upper bounds, but the quoted passage identifies the stronger analogue of the linear-complexity result as still open. Thus the unresolved issue is whether the logarithmic deviation can be reduced to a bounded one.