In random access with multi-packet reception for the 5th generation mobile communication system and beyond, the system prepares sufficient number of physical channel identifiers (PCIDs) in order to separate multiple user packet signals at the base station receiver. The base station receiver performs active user detection and channel estimation using preamble sequences, each of which has a one-to-one relationship with a PCID. As the number of PCIDs prepared in the system is increased, the computational complexity for active user detection and channel estimation becomes prohibitive. To address this problem, we propose three preamble sequence designs appropriate for random access achieving multi-packet reception. Assuming single-carrier transmission based on discrete Fourier transform (DFT)-spread orthogonal frequency division multiplexing (OFDM), the proposed methods construct the preamble sequence through repetition of the basic Zadoff-Chu (ZC) sequence, which is shorter than the length of one DFT block, with multiplication of the intra-DFT block orthogonal masking sequence. In this way, we reduce the computational complexity required for the correlation detection between the received signal and the preamble sequence. The cyclic-shifted ZC sequence-based intra-DFT block orthogonal masking mitigates the intra-DFT block interference caused by the multipath channel. Computer simulation results show quantitatively the effectiveness of the proposed methods.