Enumeration of $\mathrm{A}\mathrm{G}\mathrm{L}(\frac{m}{3},{\mathbb{F}}_{p^3})$-Invariant Extended Cyclic Codes

Let $p$ be a prime and let $r,e,m$ be positive integers such that $r\mathrm{\mid }e$ and $e\mathrm{\mid }m$. The enumeration of linear codes of length $p^m$ over ${\mathbb{F}}_{p^r}$ which are invariant under the affine linear group ${\rm AGL}(\frac me, {\Bbb F}_{p^e})$ is equivalent to the enumeration of certain ideals in a partially ordered set $(\mathcal{U},\prec )$ where ${\mathcal U}={0,1,…,\frac me(p-1)}^e$and$\prec$isdefinedbyan$e$-dimensionalsimplicialcone\mathrm{.}When$e=2$, the enumeration problem was solved in an earlier paper. In the present paper, we consider the cases $e=3$. We describe methods for enumerating all $\mathrm{A}\mathrm{G}\mathrm{L}(\frac{m}{3},{\mathbb{F}}_{p^3})$-invariant linear codes of length $p^m$ over ${\mathbb{F}}_{p^r}$