We investigate the peak-to-average power ratio (PAPR) reduction problem in orthogonal frequency-division multiplexing-based massive multiuser (MU) multiple-input multiple-output (MIMO) downlink systems. In this paper, we develop a downlink transmission scheme that performs jointly MU precoding and PAPR reduction (PP) by exploiting the excess degrees of freedom offered by equipping the BS by a large number of antennas. Specifically, the joint MU precoding and PAPR reduction scheme is formulated as a simple convex optimization problem solved via steepest gradient descent (GD) approach. Then, we develop a novel algorithm, referred to as MU-PP-GDm, to reduce the PAPR of the transmitted signals by exploiting the high-dimensional null space of the MIMO channel matrix while maintaining an excellent transmission quality. The simulation results show that the proposed MU-PP-GDm has low computational complexity and can achieve substantial PAPR performance with a fast convergence rate. INDEX TERMS 5G+, massive multiple-input multiple-output (MIMO), orthogonal frequency-division multiplexing (OFDM), peak-to-average power ratio (PAPR), multiuser (MU) precoding, convex optimization, gradient descent (GD). The associate editor coordinating the review of this manuscript and approving it for publication was Jiayi Zhang. In practice, broadband wireless communications, however, encounter large delay spread, and, therefore, suffer from frequency-selective fading. Orthogonal frequency-division multiplexing (OFDM), a scheme of encoding digital symbols on multiple orthogonal sub-carriers, is an efficient and well-established way to deal with frequency-selective channels. Therefore, massive MIMO-OFDM is a very promising combination to meet the ever growing demands for higher link readability and spectrum efficiency of next-generation wireless communication systems (5G and Beyond). However, massive MIMO precoders exhibit transmit signals with high peak-to-average power ratio (PAPR), regardless of whether single-carrier or OFDM transmission and of whether a low or a high modulation order is used [4]. Consequently, the nonlinearity of the radio frequency (RF) high-power amplifier (HPA), which is necessary in a transmission chain, yields in-band distortion and out-of-band radiation (OBR) which cause signal distortion, phase rotation and adjacent channel interference, respectively. To avoid such heavy distortions, the transmit signals require that the power amplifiers are backed-off and operated in their linear region (i.e., where their transfer characteristics are sufficiently linear). Nevertheless, operating at lower power levels reduces 25474 2169-3536 2019 IEEE. Translations and content mining are permitted for academic research only.