  In this study, we propose partitioned complementary sequences (CSs) where the
gaps between the clusters encode information bits to achieve low
peak-to-average-power ratio (PAPR) orthogonal frequency division multiplexing
(OFDM) symbols. We show that the partitioning rule without losing the feature
of being a CS coincides with the non-squashing partitions of a positive integer
and leads to a symmetric separation of clusters. We analytically derive the
number of partitioned CSs for given bandwidth and a minimum distance constraint
and obtain the corresponding recursive methods for enumerating the values of
separations. We show that partitioning can increase the spectral efficiency
(SE) without changing the alphabet of the nonzero elements of the CS, i.e.,
standard CSs relying on Reed-Muller (RM) code. We also develop an encoder for
partitioned CSs and a maximum-likelihood-based recursive decoder for additive
white Gaussian noise (AWGN) and fading channels. Our results indicate that the
partitioned CSs under a minimum distance constraint can perform similar to the
standard CSs in terms of average block error rate (BLER) and provide a higher
SE at the expense of a limited signal-to-noise ratio (SNR) loss.
