Enumeration of AGL\((\frac{m}{3}, \mathbb F_{p^3})\)-invariant extended cyclic codes (Q622785)

Summary: Let \(p\) be a prime and let \(r, e\) and \(m\) be positive integers such that \(r|e\) and \(e|m\). The enumeration of linear codes of length \(p^{m}\) over \(\mathbb F_{p^r}\) which are invariant under the affine linear group AGL\((m/e, \mathbb F_{p^e})\) is equivalent to the enumeration of certain ideals in a partially ordered set \((\mathcal U, \prec )\) where \(\mathcal U = \{0, 1, \cdots , m/e(p - 1)\}^e\) and \(\prec \) is defined by an \(e\)-dimensional simplicial cone. When \(e = 2\), the enumeration problem was solved in an earlier paper. In this paper, we consider the cases \(e = 3\). We describe methods for enumerating all AGL\((m/3, \mathbb F_{p^3})\)-invariant linear codes of length \(p^m\) over \(\mathbb F_{p^r}\).