A　quadrature　spatial　modulation　with　the　fourth　order　transmit　diversity　(FO-QSM)　scheme　is　presented　in　this　paper.　In　this　scheme,　two　dispersion-matrix　(DM)　sets　are　assigned　at　the　transmitter,　and　P　out　of　Q　DMs　are　activated　in　each　set　to　extend　the　real　and　imaginary　parts　of　P　QSM　signals　respectively,　to　large　number　of　transmit　antennas.　The　two　DM　sets　are　constructed　by　performing　Kronecker　product　between　two　simple　extension　matrices　and　the　DMs　of　Sezginer-Sari-Biglieri　(SSB)　code.　The　framework　designs　of　FO-QSM　and　the　two　DM　sets　make　FO-QSM　possess　the　properties:　1)　it　guarantees　the　fourth　order　transmit　diversity　without　requiring　any　parameter　optimization;　2)　it　has　higher　spectral　efficiency　than　the　newly　proposed　generalized　space-time　block　coded　spatial　modulation　(GSTBC-SM)　scheme.　Furthermore,　a　repeated　computation　reduced　sphere　decoding　(RCR-SD)　is　proposed.　RCR-SD　is　very　suitable　for　the　schemes　constructed　by　a　large　number　of　DMs,　including　FO-QSM.　By　sorting　the　index　vectors,　the　RCR-SD　detector　can　efficiently　reduce　the　repeated　computations　in　the　SD　searching,　which　significantly　reduces　the　decoding　complexity　without　sacrificing　optimal　decoding　performance.　Simulation　results　show　that　FO-QSM　has　obvious　better　bit　error　rate　(BER)　performance　than　GSTBC-SM.　©　1967-2012　IEEE.