Let $p$ be a prime and let $r, e, m$ be positive integers such that $r|e$ and $e|m$. The enumeration of linear codes of length $p^m$ over ${ \Bbb F}_{ p^r}$ 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 $\prec$ is defined by an $e$-dimensional simplicial cone. 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 ${ \rm AGL}(\frac m3, { \Bbb F}_{ p^3})$-invariant linear codes of length $p^m$ over ${ \Bbb F}_{ p^r}$
Categories: math.CO