This paper focuses on the age of information (AoI) under multiple access in a multiple-packet reception (MPR) channel without feedback and synchronization. Unlike traditional probabilistic schemes, we design protocol-sequences-based schemes where each user is assigned a periodic binary sequence to determine when to transmit. We use a previously known construction to generate protocol sequences that are found favorable for the AoI improvement under the MPR capability $\gamma=2$. Under this construction, by dealing with the Hamming cross-correlation property of the corresponding protocol sequences, which is more complicated than that in prior studies, we provide an analytical approach for evaluating the AoI for $\gamma=2$. When $\gamma>2$, we also propose a new construction to produce protocol sequences for the AoI improvement. Simulation results show that the proposed schemes outperform slotted ALOHA for various settings.