Good asymptotic behavior of additive cyclic codes on $ \mathbb{F}_{q}[u]‎/ ‎\langle u^{2} \rangle \times \mathbb{F}_{q}[u]‎/ ‎\langle u^{3} \rangle $

Isapour-khsadan, A.; Mehry, Sh.; Mahmoudi, S.

doi:10.22124/jart.2025.30109.1780

Good asymptotic behavior of additive cyclic codes on $ \mathbb{F}_{q}[u]‎/ ‎\langle u^{2} \rangle \times \mathbb{F}_{q}[u]‎/ ‎\langle u^{3} \rangle $

Articles in Press

Document Type : Research Paper

Authors

¹ Faculty of Mathematical Sciences and Statistics‎, ‎Malayer University, Malayer‎, ‎Iran ‎Faculty of Basic Sciences‎, ‎Khatam-ol-Anbia(PBU) University‎, ‎Tehran‎, ‎Iran

² Faculty of Mathematical Sciences and Statistics‎, ‎Malayer University, Malayer‎, ‎Iran

³ Department of Mathematics‎, ‎Bu Ali Sina University‎, ‎Hamedan‎, ‎Iran

10.22124/jart.2025.30109.1780

Abstract

‎Let $ S= \mathbb{F}_{q}[u]‎/ ‎\langle u^2\rangle$ = $ \mathbb{F}_{q}+u\mathbb{F}_{q}$ and $R=\mathbb{F}_{q}[u]‎/ ‎\langle u^3\rangle$ = $\mathbb{F}_{q}+u\mathbb{F}_{q}‎ + ‎u^{2}\mathbb{F}_{q}$ are two finite chain rings‎, ‎where $ u^{2}=0=u^{3} $ and $ q $ is a power of a prime number‎. ‎We construct a class of $ SR $-additive cyclic codes generated by pairs of polynomials‎, ‎where $ S $ is a $ R $-algebra and $ SR $-additive cyclic code is a $ R $-submodul of $ S^{\alpha} \times R^{\beta} $‎ . ‎Based on probabilistic arguments‎, ‎we study the asymptotic behaviour of the rates and relative minimum distances of a certain class of the codes‎. ‎We show that there exists an asymptotically good infinite seqence of $ SR $-additive cyclic codes with the relative minimum distance of the code is convergent to $ \delta $‎, ‎and the rat is convergent to $ \frac{2}{q+q^{2}} $ for $ 0 < \delta < \frac{1}{1+q} $‎.

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