  We study cyclotomic quiver Hecke algebras $R^{\Lambda_0}(\beta)$ in type
$A^{(2)}_{2\ell}$, where $\Lambda_0$ is the fundamental weight. The algebras
are natural $A^{(2)}_{2\ell}$-type analogue of Iwahori-Hecke algebras
associated with the symmetric group, from the viewpoint of the Fock space
theory developed by the first author and his collaborators. We give a formula
for the dimension of the algebra, and a simple criterion to tell the
representation type. The criterion is a natural generalization of Erdmann and
Nakano's for the Iwahori-Hecke algebras. Except for the examples coming from
cyclotomic Hecke algebras, no results of these kind existed for cyclotomic
quiver Hecke algebras, and our results are the first instances beyond the case
of cyclotomic Hecke algebras.
