The $2$-adic complexity has been well-analyzed in the periodic case. However, we are not aware of any theoretical results on the $N$th $2$-adic complexity of any promising candidate for a pseudorandom sequence of finite length $N$ or results on a part of the period of length $N$ of a periodic sequence, respectively. Here we introduce the first method for this aperiodic case. More precisely, we study the relation between $N$th maximum-order complexity and $N$th $2$-adic complexity of binary sequences and prove a lower bound on the $N$th $2$-adic complexity in terms of the $N$th maximum-order complexity. Then any known lower bound on the $N$th maximum-order complexity implies a lower bound on the $N$th $2$-adic complexity of the same order of magnitude. In the periodic case, one can prove a slightly better result. The latter bound is sharp which is illustrated by the maximum-order complexity of $\ell$-sequences. The idea of the proof helps us to characterize the maximum-order complexity of periodic sequences in terms of the unique rational number defined by the sequence. We also show that a periodic sequence of maximal maximum-order complexity must be also of maximal $2$-adic complexity.

翻译：$2$-进制复杂度在周期情形下已得到充分分析。然而，目前尚未见到关于有限长度$N$的伪随机序列候选者的$N$阶$2$-进制复杂度的理论结果，或关于周期序列长度为$N$的部分周期上的相应结果。本文首次提出了针对该非周期情形的方法。具体而言，我们研究了二元序列的$N$阶最大阶复杂度与$N$阶$2$-进制复杂度之间的关系，并基于$N$阶最大阶复杂度证明了$N$阶$2$-进制复杂度的一个下界。由此，任何已知的$N$阶最大阶复杂度下界都将导出同量级的$N$阶$2$-进制复杂度下界。在周期情形下，可证明一个略优的结果。该下界是紧的，这一点通过$\ell$-序列的最大阶复杂度得以说明。证明思路有助于我们利用序列定义的唯一有理数来刻画周期序列的最大阶复杂度。我们还证明：具有最大最大阶复杂度的周期序列必然也具有最大$2$-进制复杂度。