In this paper, we design space-time block codes (STBCs) to broadcast the common information omnidirectionally in a massive MIMO downlink. To reduce the burden of the downlink channel estimation and achieve partial spatial diversity from base station (BS) transmit antennas, we propose channel-independently precoded low-dimensional STBC. The precoding matrix and the signal constellation in the low-dimensional STBC are jointly designed to guarantee omnidirectional coverage at any instant time and sufficiently utilize the power amplifier capacities of BS transmit antennas, and at the same time, achieve the full diversity of the low-dimensional STBC. Under this framework, several designs are presented. To provide transmit diversity order of two, a precoded Alamouti code is proposed, which has a fast symbol-wise maximum-likelihood (ML) decoding. To provide transmit diversity order of four, three types of STBCs are proposed, being referred to as precoded orthogonal STBC (OSTBC), precoded quasi-orthogonal STBC (QOSTBC), and precoded coordinate interleaved orthogonal design (CIOD), respectively. The last two codes have the same complexity for pair-wise ML decoding, while precoded QOSTBC has a higher coding gain when the bit rate is lower than or equal to 4 bps/Hz, and precoded CIOD has a higher coding gain when the bit rate is higher than 4 bps/Hz. Precoded OSTBC has a higher decoding complexity and a lower coding gain than the other two codes, since in the precoded OSTBC the information symbols need to be jointly designed and decoded. Moreover, a precoded no-zero-entry Toeplitz code and a precoded no-zero-entry overlapped Alamouti code are also proposed. These two codes can achieve a higher diversity order with linear receivers.