$ \mathbb{Z}_{p^r}\mathbb{Z}_{p^s}\mathbb{Z}_{p^t} $-additive cyclic codes

Raziyeh Molaei; Kazem Khashyarmanesh

doi:10.3934/amc.2022079

Article Contents

2024, Volume 18, Issue 5: 1195-1215. Doi: 10.3934/amc.2022079

This issue Previous Article Post-quantum secure fully-dynamic logarithmic-size deniable group signature in code-based setting Next Article Concrete analysis of approximate ideal-SIVP to decision ring-LWE reduction

$ \mathbb{Z}_{p^r}\mathbb{Z}_{p^s}\mathbb{Z}_{p^t} $-additive cyclic codes

Raziyeh Molaei^1, and
Kazem Khashyarmanesh^2, ,

1.
Department of Pure Mathematics, Ferdowsi University of Mashhad, P.O.Box 1159-91775, Mashhad, Iran
2.
Department of Pure Mathematics, Center of Excellence in Analysis on Algebraic Structures (CEAAS), Ferdowsi University of Mashhad, P.O.Box 1159-91775, Mashhad, Iran

^*Corresponding author: Kazem Khashyarmanesh
^*Corresponding author: Kazem Khashyarmanesh

Received: March 2022

Revised: July 2022

Early access: September 15, 2022

Published online: September 15, 2022

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Abstract

Let $ p $ be a prime number and $ r, s, t $ be positive integers such that $ r\le s\le t $. A $ \mathbb{Z}_{p^r}\mathbb{Z}_{p^s}\mathbb{Z}_{p^t} $-additive code is a $ \mathbb{Z}_{p^t} $-submodule of $ \mathbb{Z}_{p^r}^{\alpha} \times \mathbb{Z}_{p^s}^{\beta} \times \mathbb{Z}_{p^t}^{\gamma} $, where $ \alpha, \beta, \gamma $ are positive integers. In this paper, we study $ \mathbb{Z}_{p^r}\mathbb{Z}_{p^s}\mathbb{Z}_{p^t} $-additive cyclic codes. In fact, we show that these codes can be identified as submodules of the ring $ R = \mathbb{Z}_{p^r}[x]/\big<x^\alpha-1\big> \times \mathbb{Z}_{p^s}[x]/\big<x^\beta-1\big> \times \mathbb{Z}_{p^t}[x]/\big<x^\gamma-1\big> $. Furthermore, we determine the generator polynomials and minimum generating sets of this kind of codes. Moreover, we investigate their dual codes.

Keywords:

$ \mathbb{Z}_{p^r}\mathbb{Z}_{p^s}\mathbb{Z}_{p^t} $-cyclic codes,
generator polynomials,
minimum generating sets,
dual code.

Mathematics Subject Classification: Primary: 12E20, 94B05; Secondary: 94B15, 94B60.

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