This paper develops a method for constructing an OFDM signal from a pair of complementary sequences so that the resulting signal is constant-modulus. A recursive method of constructing complementary sequences is developed such that the length of the signal grows linearly with the number of information symbols encoded. The constant-modulus property is exploited at each stage of the backwards symbol-decoding iteration through simple means to progressively reduce noise. For diversity comparing purposes, the new scheme is compared in symbol and frame error rates to m-PSK transmission with time-diversity in AWGN. The constant-modulus signal construction is then extended to two dimensions for the motivating application where we expect this work to have the greatest impact: massive MIMO at mmWave frequencies.