Double circulant codes for the Lee and Euclidean distance

This paper investigates double circulant codes of length $2n$ over ${\mathbb{Z}}_{p^m}$ where $p$ is an odd prime, $n$ goes to infinity, and $m\ge 1$ is a fixed integer. Using random coding, we obtain families of asymptotically good Lee codes over ${\mathbb{Z}}_{p^m}$ in the case of small and large alphabets, and asymptotically good Euclidean codes over ${\mathbb{Z}}_{p^m}$ for small alphabets. We use Euclidean codes to construct spherical codes, and Lee codes to construct insertion/deletion codes, by a projection technique due to (Yaglom, 1958) for spherical codes, and to (Sok et al., 2018) for deletion codes.