  In this work we present an algorithm to identify compact groups (CGs) that
closely follows Hickson's original aim and that improves the completeness of
the samples of compact groups obtained from redshift surveys. Instead of
identifying CGs in projection first and then checking a velocity concordance
criterion, we identify them directly in redshift space using Hickson-like
criteria. The methodology was tested on a mock lightcone of galaxies built from
the outputs of a recent semi-analytic model of galaxy formation run on top of
the Millennium Simulation I after scaling to represent the first-year Planck
cosmology. The new algorithm identifies nearly twice as many CGs, no longer
missing CGs that failed the isolation criterion because of velocity outliers
lying in the isolation annulus. The new CG sample picks up lower surface
brightness groups, which are both looser and with fainter brightest galaxies,
missed by the classic method. A new catalogue of compact groups from the Sloan
Digital Sky Survey is the natural corollary of this study. The publicly
available sample comprises $462$ observational groups with four or more galaxy
members, of which $406$ clearly fulfil all the compact group requirements:
compactness, isolation, and velocity concordance of all of their members. The
remaining $56$ groups need further redshift information of potentially
contaminating sources. This constitutes the largest sample of groups that
strictly satisfy all the Hickson's criteria in a survey with available
spectroscopic information.
