doi: 10.3934/amc.2022023
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$\mathbb{F}_{p^{m}}\mathbb{F}_{p^{m}}{[u^2]}$-additive skew cyclic codes of length $2p^s $

1. 

Department of Mathematics, Malayer University, Malayer, Iran

2. 

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

*Corresponding author: Rashid Rezaei

Received  September 2021 Revised  January 2022 Early access March 2022

In this paper, we first study the skew cyclic codes of length $ p^s $ over $ R_3 = \mathbb{F}_{p^m}+u\mathbb{F}_{p^m}+u^2\mathbb{F}_{p^m}, $ where $ p $ is a prime number and $ u^3 = 0. $ Then we characterize the algebraic structure of $ \mathbb{F}_{p^{m}}\mathbb{F}_{p^{m}}[u^2] $-additive skew cyclic codes of length $ 2p^s. $ We will show that there are sixteen different types of these codes and classify them in terms of their generators.

Citation: Roghayeh Mohammadi Hesari, Mahboubeh Hosseinabadi, Rashid Rezaei, Karim Samei. $\mathbb{F}_{p^{m}}\mathbb{F}_{p^{m}}{[u^2]}$-additive skew cyclic codes of length $2p^s $. Advances in Mathematics of Communications, doi: 10.3934/amc.2022023
References:
[1]

Y. Al-Khamees, The determination of the group of automorphisms of a finite chain ring of characteristic $p$, Quart. J. Math. Oxford Ser. (2), 42 (1991), 387-391.  doi: 10.1093/qmath/42.1.387.

[2]

I. AydogduT. Abualrub and I. Siap, On $\mathbb{Z}_2\mathbb{Z}_2[u]$-additive codes, Int. J. Comput. Math., 92 (2015), 1806-1814.  doi: 10.1080/00207160.2013.859854.

[3]

I. AydogduT. AbualrubI. Siap and N. Aydin, $\mathbb{Z}_2\mathbb{Z}_2[u]$-cyclic and constacyclic codes, IEEE. Trans. Inform. Theory, 63 (2017), 4883-4893.  doi: 10.1109/TIT.2016.2632163.

[4]

I. AydogduI. Siap and R. Ten-Valls, On the structure of $\mathbb{Z}_2\mathbb{Z}_2[u^3]$-linear and cyclic codes, Finite Fields Appl., 48 (2017), 241-260.  doi: 10.1016/j.ffa.2017.03.001.

[5]

N. Benbelkacem, M. F. Ezerman, T. Abualrub, N. Aydin and A. Batoul, Skew cyclic codes over $\mathbb{F}_4R, $, J. Algebra Appl., 21 (2022), 20pp. doi: 10.1142/S0219498822500657.

[6]

D. Boucher, W. Geiselmann and F. Ulmer, Skew-cyclic codes, Appl. Algebra Engrg. Comm. Comput., 18 (2007) 379–389. doi: 10.1007/s00200-007-0043-z.

[7]

D. BoucherP. Solé and F. Ulmer, Skew constacyclic codes over Galois rings, Adv. Math. Commun., 2 (2008), 273-292.  doi: 10.3934/amc.2008.2.273.

[8]

D. Boucher and F. Ulmer, A note on the dual codes of module skew codes, in Cryptography and Coding, Lecture Notes in Comput. Sci., 7089, 2011, Springer, Heidelberg, 230–243. doi: 10.1007/978-3-642-25516-8_14.

[9]

D. Boucher and F. Ulmer, Self-dual skew codes and factorization of skew polynomials, J. Symbolic Comput., 60 (2014), 47-61.  doi: 10.1016/j.jsc.2013.10.003.

[10]

S. M. Dodunekov and I. N. Landjev, Near-MDS codes over some small fields, Discrete Math., 213 (2000), 55-65.  doi: 10.1016/S0012-365X(99)00168-5.

[11]

R. M. Hesari, R. Rezaei and K. Samei, On self-dual skew cyclic codes of length $p^s$ over $ \mathbb{F}_{p^m} + u\mathbb{F}_{p^m}$, Discrete Math., 344 (2021), 16pp. doi: 10.1016/j.disc.2021.112569.

[12]

S. JitmanS. Ling and P. Udomkavanich, Skew constacyclic codes over finite chain rings, Adv. Math. Commun., 6 (2012), 39-63.  doi: 10.3934/amc.2012.6.39.

[13]

X. Liu and X. Xu, Some classes of repeated-root constacyclic codes over $ \mathbb{F}_{p^m} + u\mathbb{F}_{p^m}+u^2 \mathbb{F}_{p^m}$, J. Korean Math. Soc., 51 (2014), 853-866.  doi: 10.4134/JKMS.2014.51.4.853.

[14]

S. Mahmoudi and K. Samei, $SR$-additive codes, Bull. Korean Math. Soc., 56 (2019), 1235-1255.  doi: 10.4134/BKMS.b180995.

[15]

B. R. McDonald, Finite Rings with Identity, Pure and Applied Mathematics, 28, Marcel Dekker, Inc., New York, 1974.

[16]

A. Sharma and M. Bhaintwal, $ \mathbb{F}_3 R$-skew cyclic codes, Int. J. Inf. Coding Theory, 3 (2016), 234-251.  doi: 10.1504/IJICOT.2016.076967.

[17]

R. Sobhani, Complete classification of $ (\delta+\alpha u^2)$-constacyclic codes of length $ p^k$ over $ \mathbb{F}_{p^m} + u\mathbb{F}_{p^m}+u^2 \mathbb{F}_{p^m}$, Finite Fields Appl., 34 (2015), 123-138.  doi: 10.1016/j.ffa.2015.01.008.

show all references

References:
[1]

Y. Al-Khamees, The determination of the group of automorphisms of a finite chain ring of characteristic $p$, Quart. J. Math. Oxford Ser. (2), 42 (1991), 387-391.  doi: 10.1093/qmath/42.1.387.

[2]

I. AydogduT. Abualrub and I. Siap, On $\mathbb{Z}_2\mathbb{Z}_2[u]$-additive codes, Int. J. Comput. Math., 92 (2015), 1806-1814.  doi: 10.1080/00207160.2013.859854.

[3]

I. AydogduT. AbualrubI. Siap and N. Aydin, $\mathbb{Z}_2\mathbb{Z}_2[u]$-cyclic and constacyclic codes, IEEE. Trans. Inform. Theory, 63 (2017), 4883-4893.  doi: 10.1109/TIT.2016.2632163.

[4]

I. AydogduI. Siap and R. Ten-Valls, On the structure of $\mathbb{Z}_2\mathbb{Z}_2[u^3]$-linear and cyclic codes, Finite Fields Appl., 48 (2017), 241-260.  doi: 10.1016/j.ffa.2017.03.001.

[5]

N. Benbelkacem, M. F. Ezerman, T. Abualrub, N. Aydin and A. Batoul, Skew cyclic codes over $\mathbb{F}_4R, $, J. Algebra Appl., 21 (2022), 20pp. doi: 10.1142/S0219498822500657.

[6]

D. Boucher, W. Geiselmann and F. Ulmer, Skew-cyclic codes, Appl. Algebra Engrg. Comm. Comput., 18 (2007) 379–389. doi: 10.1007/s00200-007-0043-z.

[7]

D. BoucherP. Solé and F. Ulmer, Skew constacyclic codes over Galois rings, Adv. Math. Commun., 2 (2008), 273-292.  doi: 10.3934/amc.2008.2.273.

[8]

D. Boucher and F. Ulmer, A note on the dual codes of module skew codes, in Cryptography and Coding, Lecture Notes in Comput. Sci., 7089, 2011, Springer, Heidelberg, 230–243. doi: 10.1007/978-3-642-25516-8_14.

[9]

D. Boucher and F. Ulmer, Self-dual skew codes and factorization of skew polynomials, J. Symbolic Comput., 60 (2014), 47-61.  doi: 10.1016/j.jsc.2013.10.003.

[10]

S. M. Dodunekov and I. N. Landjev, Near-MDS codes over some small fields, Discrete Math., 213 (2000), 55-65.  doi: 10.1016/S0012-365X(99)00168-5.

[11]

R. M. Hesari, R. Rezaei and K. Samei, On self-dual skew cyclic codes of length $p^s$ over $ \mathbb{F}_{p^m} + u\mathbb{F}_{p^m}$, Discrete Math., 344 (2021), 16pp. doi: 10.1016/j.disc.2021.112569.

[12]

S. JitmanS. Ling and P. Udomkavanich, Skew constacyclic codes over finite chain rings, Adv. Math. Commun., 6 (2012), 39-63.  doi: 10.3934/amc.2012.6.39.

[13]

X. Liu and X. Xu, Some classes of repeated-root constacyclic codes over $ \mathbb{F}_{p^m} + u\mathbb{F}_{p^m}+u^2 \mathbb{F}_{p^m}$, J. Korean Math. Soc., 51 (2014), 853-866.  doi: 10.4134/JKMS.2014.51.4.853.

[14]

S. Mahmoudi and K. Samei, $SR$-additive codes, Bull. Korean Math. Soc., 56 (2019), 1235-1255.  doi: 10.4134/BKMS.b180995.

[15]

B. R. McDonald, Finite Rings with Identity, Pure and Applied Mathematics, 28, Marcel Dekker, Inc., New York, 1974.

[16]

A. Sharma and M. Bhaintwal, $ \mathbb{F}_3 R$-skew cyclic codes, Int. J. Inf. Coding Theory, 3 (2016), 234-251.  doi: 10.1504/IJICOT.2016.076967.

[17]

R. Sobhani, Complete classification of $ (\delta+\alpha u^2)$-constacyclic codes of length $ p^k$ over $ \mathbb{F}_{p^m} + u\mathbb{F}_{p^m}+u^2 \mathbb{F}_{p^m}$, Finite Fields Appl., 34 (2015), 123-138.  doi: 10.1016/j.ffa.2015.01.008.

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