  We investigate combinatorial, topological and algebraic properties of certain
classes of neural codes. We look into a conjecture that states if the minimal
\textit{open convex} embedding dimension of a neural code is two then its
minimal \textit{convex} embedding dimension is also two. We prove the
conjecture for two interesting classes of examples and provide a counterexample
for the converse of the conjecture. We introduce a new class of neural codes,
\textit{Doublet maximal}. We show that a Doublet maximal code is open convex if
and only if it is max-intersection complete. We prove that surjective neural
ring homomorphisms preserve max-intersection complete property. We introduce
another class of neural codes, \textit{Circulant codes}. We give the count of
neural ring endomorphisms for several sub-classes of this class.
