In this paper, we design and investigate two subcarrier-allocation algorithms for downlink orthogonal frequency-division multiple-access (OFDMA) systems. One is designed to avoid assigning users, by as many as possible, worst subchannels, which is referred to as the bidirectional worst subchannel avoiding (BWSA) algorithm. The other one is called the best subchannel seeking (BSS) algorithm, which aims at assigning users the best possible subchannels. We study and compare the error rate, spectrum efficiency, and the complexity of the two algorithms. We also compare them with some existing subcarrier-allocation algorithms. Our studies show that both the BWSA and BSS algorithms belong to the class of low-complexity subcarrier-allocation algorithms. In general, they outperform all the other considered suboptimal algorithms in terms of both error performance and spectrum efficiency. When operated in relatively large OFDMA systems, we find that both the error performance and the spectrum efficiency attainable by the BWSA and BSS algorithms are close to those achieved by the Hungarian algorithm of high complexity.