Research Article
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Year 2019, , 39 - 51, 19.01.2019
https://doi.org/10.13069/jacodesmath.514339

Abstract

References

  • [1] T. Abualrub, I. Siap, Cyclic codes over the rings ${Z}_{2}+u{Z}_{2}$ and ${Z}_{2}+u{Z}_{2}+{u}^2{Z}_{2}$, Des. Codes Cryptogr. 42(3) (2007) 273-287.
  • [2] M. Al-Ashker, M. Hamoudeh, Cyclic codes over $Z_2+uZ_2+u^2Z_2+\ldots+u^{k-1}Z_2$, Turk J Math 35 (2011) 737-749.
  • [3] I. Aydogdu, T. Abualrub, I. Siap, On $\mathbb{Z}_{2}\mathbb{Z}_{2}[u]$-additive codes, Int. J. Comput. Math. 92(9) (2015) 1806-1814.
  • [4] I. Aydogdu, I. Siap, The structure of $\mathbb{Z}_{2}\mathbb{Z}_{2^s}$-additive codes: Bounds on the minimum distance, Appl.Math. Inf. Sci. 7(6) (2013) 2271-2278.
  • [5] I. Aydogdu, I. Siap, On $\mathbb{Z}_{p^r}\mathbb{Z}_{p^s}$-additive codes, Linear Multilinear Algebra 63(10) (2015) 2089-2102.
  • [6] A. Bonnecaze, P. Udaya, Cyclic codes and selfdual codes over ${F}_{2} + u{F}_{2}$, IEEE Trans. Inform. Theory 45(4) (1999) 1250-1255.
  • [7] J. Borges, C. Fernández-Córdoba, J. Pujol, J. Rifà, M. Villanueva, $\mathbb{Z}_{2}\mathbb{Z}_{4}$-linear codes: Generator matrices and duality, Des. Codes Cryptogr. 54(2) (2010) 167-179.
  • [8] A. R. Hammons, V. Kumar, A. R. Calderbank, N. J. A. Sloane, P. Solé, The $\mathbb{Z}_{4}$-linearity of Kerdock, Preparata, Goethals, and Related codes, IEEE Trans. Inform. Theory 40(2) (1994) 301-319.
  • [9] G. H. Norton, A. Salagean, On the structure of linear and cyclic codes over a finite chain ring, Appl.Algebra Engrg. Comm. Comput. 10(6) (2000) 489-506.

Codes over $\mathbb{Z}_{p}[u]/{\langle u^r \rangle}\times\mathbb{Z}_{p}[u]/{\langle u^s \rangle}$

Year 2019, , 39 - 51, 19.01.2019
https://doi.org/10.13069/jacodesmath.514339

Abstract

In this paper we generalize $\mathbb{Z}_{2}\mathbb{Z}_{2}[u]$-linear codes to codes over $\mathbb{Z}_{p}[u]/{\langle u^r \rangle}\times\mathbb{Z}_{p}[u]/{\langle u^s \rangle}$ where $p$ is a prime number and $u^r=0=u^s$. We will call these family of codes as $\mathbb{Z}_{p}[u^r,u^s]$-linear codes which are actually special submodules. We determine the standard forms of the generator and parity-check matrices of these codes. Furthermore, for the special case $p=2$, we define a Gray map to explore the binary images of $\mathbb{Z}_{2}[u^r,u^s]$-linear codes. Finally, we study the structure of self-dual $\mathbb{Z}_{2}[u^2,u^3]$-linear codes and present some examples.

References

  • [1] T. Abualrub, I. Siap, Cyclic codes over the rings ${Z}_{2}+u{Z}_{2}$ and ${Z}_{2}+u{Z}_{2}+{u}^2{Z}_{2}$, Des. Codes Cryptogr. 42(3) (2007) 273-287.
  • [2] M. Al-Ashker, M. Hamoudeh, Cyclic codes over $Z_2+uZ_2+u^2Z_2+\ldots+u^{k-1}Z_2$, Turk J Math 35 (2011) 737-749.
  • [3] I. Aydogdu, T. Abualrub, I. Siap, On $\mathbb{Z}_{2}\mathbb{Z}_{2}[u]$-additive codes, Int. J. Comput. Math. 92(9) (2015) 1806-1814.
  • [4] I. Aydogdu, I. Siap, The structure of $\mathbb{Z}_{2}\mathbb{Z}_{2^s}$-additive codes: Bounds on the minimum distance, Appl.Math. Inf. Sci. 7(6) (2013) 2271-2278.
  • [5] I. Aydogdu, I. Siap, On $\mathbb{Z}_{p^r}\mathbb{Z}_{p^s}$-additive codes, Linear Multilinear Algebra 63(10) (2015) 2089-2102.
  • [6] A. Bonnecaze, P. Udaya, Cyclic codes and selfdual codes over ${F}_{2} + u{F}_{2}$, IEEE Trans. Inform. Theory 45(4) (1999) 1250-1255.
  • [7] J. Borges, C. Fernández-Córdoba, J. Pujol, J. Rifà, M. Villanueva, $\mathbb{Z}_{2}\mathbb{Z}_{4}$-linear codes: Generator matrices and duality, Des. Codes Cryptogr. 54(2) (2010) 167-179.
  • [8] A. R. Hammons, V. Kumar, A. R. Calderbank, N. J. A. Sloane, P. Solé, The $\mathbb{Z}_{4}$-linearity of Kerdock, Preparata, Goethals, and Related codes, IEEE Trans. Inform. Theory 40(2) (1994) 301-319.
  • [9] G. H. Norton, A. Salagean, On the structure of linear and cyclic codes over a finite chain ring, Appl.Algebra Engrg. Comm. Comput. 10(6) (2000) 489-506.
There are 9 citations in total.

Details

Primary Language English
Subjects Engineering
Journal Section Articles
Authors

İsmail Aydogdu This is me 0000-0001-9308-4829

Publication Date January 19, 2019
Published in Issue Year 2019

Cite

APA Aydogdu, İ. (2019). Codes over $\mathbb{Z}_{p}[u]/{\langle u^r \rangle}\times\mathbb{Z}_{p}[u]/{\langle u^s \rangle}$. Journal of Algebra Combinatorics Discrete Structures and Applications, 6(1), 39-51. https://doi.org/10.13069/jacodesmath.514339
AMA Aydogdu İ. Codes over $\mathbb{Z}_{p}[u]/{\langle u^r \rangle}\times\mathbb{Z}_{p}[u]/{\langle u^s \rangle}$. Journal of Algebra Combinatorics Discrete Structures and Applications. January 2019;6(1):39-51. doi:10.13069/jacodesmath.514339
Chicago Aydogdu, İsmail. “Codes over $\mathbb{Z}_{p}[u]/{\langle u^r \rangle}\times\mathbb{Z}_{p}[u]/{\langle u^s \rangle}$”. Journal of Algebra Combinatorics Discrete Structures and Applications 6, no. 1 (January 2019): 39-51. https://doi.org/10.13069/jacodesmath.514339.
EndNote Aydogdu İ (January 1, 2019) Codes over $\mathbb{Z}_{p}[u]/{\langle u^r \rangle}\times\mathbb{Z}_{p}[u]/{\langle u^s \rangle}$. Journal of Algebra Combinatorics Discrete Structures and Applications 6 1 39–51.
IEEE İ. Aydogdu, “Codes over $\mathbb{Z}_{p}[u]/{\langle u^r \rangle}\times\mathbb{Z}_{p}[u]/{\langle u^s \rangle}$”, Journal of Algebra Combinatorics Discrete Structures and Applications, vol. 6, no. 1, pp. 39–51, 2019, doi: 10.13069/jacodesmath.514339.
ISNAD Aydogdu, İsmail. “Codes over $\mathbb{Z}_{p}[u]/{\langle u^r \rangle}\times\mathbb{Z}_{p}[u]/{\langle u^s \rangle}$”. Journal of Algebra Combinatorics Discrete Structures and Applications 6/1 (January 2019), 39-51. https://doi.org/10.13069/jacodesmath.514339.
JAMA Aydogdu İ. Codes over $\mathbb{Z}_{p}[u]/{\langle u^r \rangle}\times\mathbb{Z}_{p}[u]/{\langle u^s \rangle}$. Journal of Algebra Combinatorics Discrete Structures and Applications. 2019;6:39–51.
MLA Aydogdu, İsmail. “Codes over $\mathbb{Z}_{p}[u]/{\langle u^r \rangle}\times\mathbb{Z}_{p}[u]/{\langle u^s \rangle}$”. Journal of Algebra Combinatorics Discrete Structures and Applications, vol. 6, no. 1, 2019, pp. 39-51, doi:10.13069/jacodesmath.514339.
Vancouver Aydogdu İ. Codes over $\mathbb{Z}_{p}[u]/{\langle u^r \rangle}\times\mathbb{Z}_{p}[u]/{\langle u^s \rangle}$. Journal of Algebra Combinatorics Discrete Structures and Applications. 2019;6(1):39-51.