The photomultiplier tube (PMT) used in the water Cherenkov detector array (WCDA) of the Large High Altitude Air Shower Observatory (LHAASO) requires a good single photoelectron (SPE) spectrum and a charge dynamic range from 1 to 4000 photoelectrons. In this paper, the bases design and improvement of the photomultiplier tube R5912 are presented. The results show that at the gain of 2.6 × 106, the anode output has a good single photoelectron spectrum, and its charge non-linearity is within 5% when the number of photoelectrons (nPE) is 3500. The charge non-linearity of the 8th dynode output is within 2% when the number of nPE is 4000, which satisfies the dynamic range requirement.