Double circulant codes for the Lee and Euclidean distance

Abstract

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.

Keywords

• double circulant codes
• lee distance
• euclidean distance
• spherical codes
• deletion codes