In the present paper, we discuss the class of Type III and Type IV codes from the perspectives of neighbors. Our investigation analogously extends the results originally presented by Dougherty [8] concerning the neighbor graph of binary self-dual codes. Moreover, as an application of neighbors in invariant theory, we show that the ring of the weight enumerators of Type II code $d_{n}^{+}$ and its neighbors in arbitrary genus is finitely generated. Finally, we obtain a minimal set of generators of this ring up to the space of degree 24 and genus 3.