We aim at performing a kinematic census of young stellar objects (YSOs) in the Rho Ophiuchi F core and partially in the E core of the L1688 dark cloud. We run a proper motion program at the ESO New Technology Telescope (NTT) with the Son of ISAAC (SOFI) instrument over nine years in the near-infrared. We complemented these observations with various public image databases to enlarge the time base of observations and the field of investigation to 0.5 deg X 0.5 deg. We derived positions and proper motions for 2213 objects. From these, 607 proper motions were derived from SOFI observations with a ~1.8 mas/yr accuracy while the remaining objects were measured only from auxiliary data with a mean precision of about ~3 mas/yr. We performed a kinematic analysis of the most accurate proper motions derived in this work, which allowed us to separate cluster members from field stars and to derive the mean properties of the cluster. From the kinematic analysis we derived a list of 68 members and 14 candidate members, comprising 26 new objects with a high membership probability. These new members are generally fainter than the known ones. We measured a mean proper motion of (mu_RA*, mu_DEC)=(-8.2, -24.3)+/-0.8 mas/yr for the L1688 dark cloud. A supervised classification was applied to photometric data of members to allocate a spectral energy distribution (SED) classification to the unclassified members.} We kinematically confirmed that the 56 members that were known from previous studies of the Rho Ophiuchi F cluster and that were also part of our survey are members of the cluster, and we added 26 new members. We defined the evolutionary status of the unclassified members of the cluster. We showed that a large part (23) of these new members are probably brown dwarfs, which multiplies the number of known substellar objects in the cluster by a factor of 3.3.
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