In this letter, we derive a novel and accurate closed-form bit error rate (BER) approximation of the optical wireless communications (OWC) systems for the sub-carrier intensity modulation (SIM) employing binary phase-shift keying (BPSK) with multiple transmit and single receive apertures over strong atmospheric turbulence channels, which makes it possible to effectively investigate and predict the BER performance for various system configurations. Furthermore, we also derive a concise asymptotic BER formula to quantitatively evaluate the asymptotically achievable error performance (i.e., asymptotic diversity and combining gains) in the high signal-to-noise (SNR) regimes. Some numerical results are provided to corroborate the accuracy and effectiveness of our theoretical expressions.