In this paper, two modulation schemes based on complementary sequences (CSs) are proposed for uplink control channels in unlicensed bands. These schemes address high peak-to-average-power ratio (PAPR) under non-contiguous resource allocation in the frequency domain and reduce the maximum PAPR to 3 dB. The first scheme allows the users to transmit a small amount of uplink control information (UCI) such as acknowledgment signals and does not introduce a trade-off between PAPR and co-channel interference (CCI). The second scheme, which enables up to 21 UCI bits for a single user or 11 UCI bits for three users in an interlace, is based on a new theorem introduced in this paper. This theorem leads distinct CSs compatible with a wide variety of resource allocations while capturing the inherent relationship between CSs and Reed-Muller (RM) codes, which makes CSs more useful for practical systems. The numerical results show that the proposed schemes maintain the low-PAPR benefits without increasing the error rate for non-contiguous resource allocations in the frequency domain.
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