The light curve of the fast radio burst (FRB) 181112 is resolved into four successive pulses, and the time interval ($sim0.8$ ms) between the first and third pulses coincides with that between the second and fourth pulses, which can be interpreted as a neutron star (NS) spinning at a period of about $0.8$ ms. Although this period is shorter than the most rapidly rotating pulsar currently known ($1.4$ ms), it is typical for a simulated massive NS formed immediately after the coalescence of binary neutron stars (BNS). Therefore, a BNS merger is a good candidate for the origin of this FRB if the periodicity is real. We discuss the future implications that can be obtained if such a periodicity is detected from FRBs simultaneously with gravitational waves (GW). The remnant spin period $P_{rm rem}$ inferred from the FRB observation is unique information which is not readily obtained by current GW observations at the post-merger phase. If combined with the mass of the merger remnant $M_{rm rem}$ inferred from GW data, it would set a new constraint on the equation of state of nuclear matter. Furthermore, the post-merger quantity $P_{rm rem}/M_{rm rem}$, or the tidal deformability of the merger remnant, is closely related to the binary tidal deformability parameter $Lambda$ of NSs before they merge, and a joint FRB-GW observation will establish a new limit on $Lambda$. Thus, if $Lambda$ is also well measured by GW data, a comparison between these two will provide further insights into the nature of nuclear matter and BNS mergers.
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