  In this study, we propose a flexible construction of complementary sequences
(CSs) that can contain zero-valued elements. To derive the construction, we use
Boolean functions to represent a polynomial generated with a recursion. By
applying this representation to recursive CS constructions, we show the impact
of construction parameters such as sign, amplitude, phase rotation used in the
recursion on the elements of the synthesized CS. As a result, we extend Davis
and Jedwab's CS construction by obtaining independent functions for the
amplitude and phase of each element of the CS, and the seed sequence positions
in the CS. The proposed construction shows that a set of distinct CSs
compatible with non-contiguous resource allocations for orthogonal
frequency-division multiplexing (OFDM) and various constellations can be
synthesized systematically. It also leads to a low peak-to-mean-envelope-power
ratio (PMEPR) multiple accessing scheme in the uplink and a low-complexity
recursive decoder. We demonstrate the performance of the proposed encoder and
decoder through comprehensive simulations.
