  Bicyclic codes are a generalization of the one dimensional (1D) cyclic codes
to two dimensions (2D). Similar to the 1D case, in some cases, 2D cyclic codes
can also be constructed to guarantee a specified minimum distance. Many aspects
of these codes are yet unexplored. Motivated by the problem of constructing
quantum codes, in this paper, we study some structural properties of certain
bicyclic codes. We show that a primitive narrow-sense bicyclic hyperbolic code
of length $n^2$ contains its dual if and only if its design distance is lower
than $n-\Delta$, where $\Delta=\mathcal{O}(\sqrt{n})$. We extend the
sufficiency condition to the non-primitive case as well. We also show that over
quadratic extension fields, a primitive bicyclic hyperbolic code of length
$n^2$ contains Hermitian dual if and only if its design distance is lower than
$n-\Delta_h$, where $\Delta_h=\mathcal{O}(\sqrt{n})$. Our results are analogous
to some structural results known for BCH and Reed-Solomon codes. They further
our understanding of bicyclic codes. We also give an application of these
results by showing that we can construct two classes of quantum bicyclic codes
based on our results.
