We describe the broadband injection-locking performance of a Fabry-Perot laser diode (FP-LD) for digital coherent transmission. The dynamic locking bandwidth of the FP-LD was as wide as 28.8 GHz, which makes it possible to achieve precise carrier-phase synchronization with extremely low phase noise over a wide frequency range. By applying the FP-LD as an LO in an injection-locked homodyne receiver for digital coherent quadrature amplitude modulation (QAM) transmission, we demonstrate, for the first time, the precise demodulation of 3, 10 and 20 Gbaud 256 QAM signals even when using a widely and randomly phase-modulated transmitter laser. This is attributed to the excellent wideband dynamic injection-locking characteristics of the FP-LD.