INTRODUCING TWO CLASSES OF OPTIMAL CODES DERIVED FROM ONE WEIGHT F qFq[u]-ADDITIVE CODES
Subject Areas : Algebraic Structures and Optimization
sadegh sadeghi
1
(
گروه ریاضی مرکز فارسان، واحد شهرکرد، دانشگاه آزاد اسلامی، شهرکرد، ایران
)
narjes mohsenifar
2
(
Department of Electrical Engineering, Shahrekord Branch, Islamic Azad University, Shahrekord, Iran
)
Keywords: Additive code, constacyclic code, one weight code. ,
Abstract :
Let $\mathbb{F}_{q}$ be a finite field with $q$ elements where $q = p^{m}$, and $R=\mathbb{F}_{q}+u \mathbb{F}_{q}$ denotes the ring $\frac{\mathbb{F}_{q}[u] }{\langle u^{2}\rangle}$. For positive integers $\alpha$ and $\beta$, a nonempty subset $C$ of $\mathbb{F}_{q}^{\alpha}\times R^{\beta}$ is called an $\mathbb{F}_{q}\mathbb{F}_{q}[u]$-additive code if $C$ is an $R$-submodule of $\mathbb{F}_{q}^{\alpha}\times R^{\beta}$. In this paper, we study these codes with respect to homogenous and Lee weights. Among main results, by the Gray image of these codes, we obtain $[q^{2}+q, 2, q^{2}]$ and $[2(q+1), 2, 2q]$ one weight optimal codes over $\mathbb{F}_{q}$.
[1] T. Abualrub, I. Siap, N. Aydin, Z2Z4-additive cyclic codes, IEEE Trans. Inform. Theory. 60 (3) (2014) 1508-1514.
[2] I . Aydogdu, T. Abualrub, I. Siap, On Z2Z2[u]-additive codes, Int. J. Inf. comput. math. 92 (9)(2015) 1806-1814.
[3] M. Bilal, J. Borges, S. T. Dougherty, C. Fern´andez-C´ordoba, Maximum distance separable codes over Z4 and Z2 × Z4, Des. Codes Cryptogr. 61 (2011) 31-40.
[4] J. Borges, C. Fern´andez-C´ordoba, There is exactly one Z2Z4-cyclic 1-perfect code, Des. Codes Cryptogr. (2016).
[5] J. Borges, C. Fern´andez-C´ordoba, J. Pujol, J. Rifa and M. Villanueva, Z2Z4-linear codes: generator matrices and duality, Des. Codes Cryptogr. 54 (2010) 167-179.
[6] B. Chen, H. Q. Dinh, H. Liu, L. Wang, Constacyclic codes of length 2ps over Fpm + uFpm, Finite fields Appl. 37 (2016) 108-130.
[7] B. Chen , Y. Fan, L. Lin, H. Liu. Constacyclic codes over finite fields. Finite fields Appl. 18 (2012) 1217-1231.
[8]
H. Q. Dinh, Constacyclic codes of length ps over Fpm + uFpm, J. Algebra. 324 (2010) 940-950. [9] H. Q. Dinh, S. Dhompongsa, S. Sriboonchitta, On constacyclic codes of length 4ps over Fpm + uF pm, Discrete Math. (2016), http://dx.doi.org/10.1016/j.disc.2016.11.014
. [10] H. Q. Dinh, B. T. Nguyen, S. Sriboonchitta, T. M. Vo, On (α + uβ)- constacyclic codes of length 4ps over Fpm + uFpm, J. Algebra Appl. (2019), http://dx.doi.org/10.1142/S0219498819500233
. [11]
S. T. Dougherty, H. Liu, L. Yu, One weight Z2Z4-additive codes, Appl. Algebra Eng. Commun. Comput. 27 (2016) 123-138. [12] K. Guenda, T. A. Gulliver, MDS and self-dual codes over rings, Finite fields Appl. 18 (2012) 1061-1075
. [13] W. C. Huffman and V. Pless, Fundamentals of Error-Correcting Codes, Cambridge University Press, Cambridge, 2003
. [14] J. Jitman, P. Udomkavanich. The Gray image of codes over finite chain rings. International Journal of Contemporary Mathematical Sciences, 5(10) (2010) 449-458
. [15] Z. Lu, S. Zhu, L. Wang, X. Kai, One-Lee weight and two-Lee weight Z2Z2[u]-additive codes.
(2016), http://arxiv.org/abs/1609.09588v1. [16] L. Qian, X. Cao, Bounds and optimal q-ary codes derived from the ZqR-cyclic codes, IEEE Trans. Inform. Theory. (2019),http://dx.doi.org/10.1109/TIT.2019.2927034
. [17] K. Samei and S. Mahmoudi, Singlton bounds for R-additive codes, Adv. Math. Commun. 12
(2018) 107-114. [18] A. Sharma, S. Rani. On constacyclic codes over finite fields. Cryptogr. Commun. (2015), http://dx.doi.org/10.1007/s12095-015-0163-4
. [19] B. Srinivasulu, M. Bhaintwal, Z2(Z2 + uZ2)-additive cyclic codes and their duals, Discrete Math. Alg. and Appl. 8 (2)(2016) 1-19
.