We generated gain-switched pulses via electrical pulse excitations in a 1270 nm distributed feedback (DFB) laser diode (LD) with a direct-modulation bandwidth of 30 GHz. The measurements revealed short-pulse widths of 5.3 and 8.8 ps with and without chirp compensation, via a single-mode optical fiber. The 5.3 ps pulses exhibited a spectral width of 0.40 nm (spectral bandwidth of 71 GHz), yielding a time-bandwidth product of 0.38. Although the gain-switched pulses in DFB LDs inherently contain linear and nonlinear chirp, optimized pumping conditions enable generation of nearly transform-limited ps pulses after linear chirp compensation.