Year 2019, Volume 6 , Issue 1, Pages 39 - 51 2019-01-19

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

Ismail Aydogdu [1]


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.
Linear codes, Self-dual codes, $\mathbb{Z}_{2}\mathbb{Z}_{2}(u)$-linear codes, $\mathbb{Z}_{p}(u^ru^s)$-linear codes
  • [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.
Primary Language en
Subjects Engineering
Journal Section Articles
Authors

Orcid: 0000-0001-9308-4829
Author: Ismail Aydogdu

Dates

Publication Date : January 19, 2019

Bibtex @research article { jacodesmath514339, journal = {Journal of Algebra Combinatorics Discrete Structures and Applications}, issn = {}, eissn = {2148-838X}, address = {}, publisher = {Yildiz Technical University}, year = {2019}, volume = {6}, pages = {39 - 51}, doi = {10.13069/jacodesmath.514339}, title = {Codes over \$\\mathbb\{Z\}\_\{p\}[u]/\{\\langle u\^r \\rangle\}\\times\\mathbb\{Z\}\_\{p\}[u]/\{\\langle u\^s \\rangle\}\$}, key = {cite}, author = {Aydogdu, Ismail} }
APA Aydogdu, I . (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 . DOI: 10.13069/jacodesmath.514339
MLA Aydogdu, I . "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 (2019 ): 39-51 <https://dergipark.org.tr/en/pub/jacodesmath/issue/42703/514339>
Chicago Aydogdu, I . "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 (2019 ): 39-51
RIS TY - JOUR T1 - Codes over $\mathbb{Z}_{p}[u]/{\langle u^r \rangle}\times\mathbb{Z}_{p}[u]/{\langle u^s \rangle}$ AU - Ismail Aydogdu Y1 - 2019 PY - 2019 N1 - doi: 10.13069/jacodesmath.514339 DO - 10.13069/jacodesmath.514339 T2 - Journal of Algebra Combinatorics Discrete Structures and Applications JF - Journal JO - JOR SP - 39 EP - 51 VL - 6 IS - 1 SN - -2148-838X M3 - doi: 10.13069/jacodesmath.514339 UR - https://doi.org/10.13069/jacodesmath.514339 Y2 - 2018 ER -
EndNote %0 Journal of Algebra Combinatorics Discrete Structures and Applications Codes over $\mathbb{Z}_{p}[u]/{\langle u^r \rangle}\times\mathbb{Z}_{p}[u]/{\langle u^s \rangle}$ %A Ismail Aydogdu %T Codes over $\mathbb{Z}_{p}[u]/{\langle u^r \rangle}\times\mathbb{Z}_{p}[u]/{\langle u^s \rangle}$ %D 2019 %J Journal of Algebra Combinatorics Discrete Structures and Applications %P -2148-838X %V 6 %N 1 %R doi: 10.13069/jacodesmath.514339 %U 10.13069/jacodesmath.514339
ISNAD Aydogdu, Ismail . "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
AMA Aydogdu I . 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.
Vancouver Aydogdu I . 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): 51-39.