Enumeration of ${\rm AGL}(\frac m3, {\Bbb F}_{p^3})$-Invariant Extended Cyclic Codes

TL;DR

This paper extends the enumeration of affine linear group-invariant codes from the case e=2 to e=3, providing methods to count certain invariant linear codes over finite fields.

Contribution

It introduces new methods for enumerating ${ m AGL}(rac m3, {F}_{p^3})$-invariant codes, advancing previous work that only addressed the case e=2.

Findings

Enumeration methods for e=3 case developed.

Connection between codes and ideals in a poset established.

Provides explicit enumeration techniques for invariant codes.

Abstract

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 ${\Bbb F}_{p^r}$