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

#### İsmail 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
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Primary Language en Engineering Articles Orcid: 0000-0001-9308-4829Author: İsmail AYDOGDU 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, İsmail} } 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 . DOI: 10.13069/jacodesmath.514339 MLA 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 6 (2019 ): 39-51 Chicago 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 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 - İsmail 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 İsmail 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, 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 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. 2019; 6(1): 39-51. 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.

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