Linear Complementary Dual Codes over the $ p $-adic Integers

Steven T. Dougherty; Esengül Saltürk

doi:10.3934/amc.2026009

Article Contents

2026, Volume 22: 217-232. Doi: 10.3934/amc.2026009

This volume Previous Article A fast multiplication algorithm and RLWE–PLWE equivalence for the maximal real subfield of the $ 2^r p^s $-th cyclotomic field Next Article The differential uniformity of three classes of power functions

Linear Complementary Dual Codes over the $ p $-adic Integers

Steven T. Dougherty^1, and
Esengül Saltürk^2, ,

1.
Department of Mathematics, University of Scranton, Scranton, PA 18510, USA
2.
Department of Computer Engineering, Atlas University, Istanbul, Turkey

^*Corresponding author: Esengül Saltürk
^*Corresponding author: Esengül Saltürk

Received: July 31, 2024

Revised: March 27, 2025

Early access: December 24, 2025

Published online: December 24, 2025

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Abstract

Linear complementary dual (LCD) codes are studied over the infinite integral domain of $ p $-adic integers. We give necessary and sufficient conditions for the existence of LCD codes in this space and describe the structure of these codes in terms of type and in terms of linear algebra over the $ p $-adic integers. We show that there exists an LCD code over $ \mathbb{Z}_{p^\infty} $ of type $ 1^k $ if and only if there exists LCD codes of type $ \{(p^{m_0})^{k_0}, (p^{m_1})^{k_1}, \ldots, (p^{m_r})^{k_r}\} $ for all $ k_0, k_1, \dots, k_r $ with $ \sum_{i = 0}^r k_i = k. $ We generalize various constructions of LCD codes over $ \mathbb{Z}_{p^\infty}. $ Finally, we study projections and lifts of these LCD codes to codes over finite chain rings. We show that if $ C $ is a code over $ \mathbb{Z}_{p^\infty} $ of type $ 1^k $ such that its projection to $ \mathbb{Z}_{p^e} $ is an LCD code, then $ C $ is an LCD code, and if $ C $ is an LCD code over $ \mathbb{Z}_{p^\infty} $ that is not of type $ 1^k $, then for infinitely many $ e $, we have that its projection to $ \mathbb{Z}_{p^e} $ is not an LCD code.

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Mathematics Subject Classification: Primary: 94B60.

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