  Bound states in the continuum (BICs) are trapped or guided modes with their
frequencies in the frequency intervals of the radiation modes. On periodic
structures, a BIC is surrounded by a family of resonant modes with their
quality factors approaching infinity. Typically the quality factors are
proportional to $1/|\beta - \beta_*|^2$, where $\beta$ and $\beta_*$ are the
Bloch wavevectors of the resonant modes and the BIC, respectively. But for some
special BICs, the quality factors are proportional to $1/|\beta -\beta_*|^4$.
In this paper, a general condition is derived for such special BICs on
two-dimensional periodic structures. As a numerical example, we use the general
condition to calculate special BICs, which are antisymmetric standing waves, on
a periodic array of circular cylinders, and show their dependence on
parameters. The special BICs are important for practical applications, because
they produce resonances with large quality factors for a very large range of
$\beta$.
