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In massive grant-free transmission, joint user activity detection (UAD) and channel estimation (CE) is essential for data recovery at the receiver, which has been extensively researched in frequency-flat fading scenarios. However, in practical orthogonal frequency division multiplexing (OFDM)-based systems, frequency-selective fading leads to a significant increase in the number of channel coefficients to be estimated, imposing new challenges for the design of joint UAD and CE algorithms. Therefore, this paper investigates joint UAD and CE for OFDM-based massive grant-free transmission over frequency-selective fading channels. Firstly, the discrete cosine transform (DCT) is employed to reformulate the compressed sensing (CS) problem with the reduced dimension of the DCT domain channel response vector. Then, we develop a hybrid message passing (HMP) algorithm under the framework of the constrained …