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.
Citation: Adel Alahmadi, Altaf Alshuhail, Alaa Altassan, Hatoon Shoaib, Patrick Solé. Double circulant codes for the Lee and Euclidean distance[J]. AIMS Mathematics, 2023, 8(10): 23566-23577. doi: 10.3934/math.20231198
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.
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Adel Alahmadi, Altaf Alshuhail, Alaa Altassan, Hatoon Shoaib, Patrick Solé. Double circulant codes for the Lee and Euclidean distance[J]. AIMS Mathematics, 2023, 8(10): 23566-23577. doi: 10.3934/math.20231198
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