Year 2019, Volume 6 , Issue 2, Pages 75 - 94 2019-05-07

Self-dual and complementary dual abelian codes over Galois rings

Somphong JİTMAN [1] , San LİNG [2]


Self-dual and complementary dual cyclic/abelian codes over finite fields form important classes of linear codes that have been extensively studied due to their rich algebraic structures and wide applications. In this paper, abelian codes over Galois rings are studied in terms of the ideals in the group ring ${ GR}(p^r,s)[G]$, where $G$ is a finite abelian group and ${ GR}(p^r,s)$ is a Galois ring. Characterizations of self-dual abelian codes have been given together with necessary and sufficient conditions for the existence of a self-dual abelian code in ${ GR}(p^r,s)[G]$. A general formula for the number of such self-dual codes is established. In the case where $\gcd(|G|,p)=1$, the number of self-dual abelian codes in ${ GR}(p^r,s)[G]$ is completely and explicitly determined. Applying known results on cyclic codes of length $p^a$ over ${ GR}(p^2,s)$, an explicit formula for the number of self-dual abelian codes in ${ GR}(p^2,s)[G]$ are given, where the Sylow $p$-subgroup of $G$ is cyclic. Subsequently, the characterization and enumeration of complementary dual abelian codes in ${ GR}(p^r,s)[G]$ are established. The analogous results for self-dual and complementary dual cyclic codes over Galois rings are therefore obtained as corollaries.
Abelian codes, Galois rings, Self-dual codes, Complementary dual codes, Codes over rings
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Primary Language en
Subjects Engineering
Journal Section Articles
Authors

Orcid: 0000-0003-1076-0866
Author: Somphong JİTMAN (Primary Author)

Orcid: 0000-0002-1978-3557
Author: San LİNG

Thanks S. Jitman was supported by the Thailand Research Fund and Silpakorn University under Research Grant RSA6280042. S. Ling was supported by Nanyang Technological University Research Grant M4080456.
Dates

Publication Date : May 7, 2019

Bibtex @research article { jacodesmath560406, journal = {Journal of Algebra Combinatorics Discrete Structures and Applications}, issn = {}, eissn = {2148-838X}, address = {}, publisher = {Yildiz Technical University}, year = {2019}, volume = {6}, pages = {75 - 94}, doi = {10.13069/jacodesmath.560406}, title = {Self-dual and complementary dual abelian codes over Galois rings}, key = {cite}, author = {Ji̇tman, Somphong and Li̇ng, San} }
APA Ji̇tman, S , Li̇ng, S . (2019). Self-dual and complementary dual abelian codes over Galois rings . Journal of Algebra Combinatorics Discrete Structures and Applications , 6 (2) , 75-94 . DOI: 10.13069/jacodesmath.560406
MLA Ji̇tman, S , Li̇ng, S . "Self-dual and complementary dual abelian codes over Galois rings" . Journal of Algebra Combinatorics Discrete Structures and Applications 6 (2019 ): 75-94 <https://dergipark.org.tr/en/pub/jacodesmath/issue/45030/560406>
Chicago Ji̇tman, S , Li̇ng, S . "Self-dual and complementary dual abelian codes over Galois rings". Journal of Algebra Combinatorics Discrete Structures and Applications 6 (2019 ): 75-94
RIS TY - JOUR T1 - Self-dual and complementary dual abelian codes over Galois rings AU - Somphong Ji̇tman , San Li̇ng Y1 - 2019 PY - 2019 N1 - doi: 10.13069/jacodesmath.560406 DO - 10.13069/jacodesmath.560406 T2 - Journal of Algebra Combinatorics Discrete Structures and Applications JF - Journal JO - JOR SP - 75 EP - 94 VL - 6 IS - 2 SN - -2148-838X M3 - doi: 10.13069/jacodesmath.560406 UR - https://doi.org/10.13069/jacodesmath.560406 Y2 - 2019 ER -
EndNote %0 Journal of Algebra Combinatorics Discrete Structures and Applications Self-dual and complementary dual abelian codes over Galois rings %A Somphong Ji̇tman , San Li̇ng %T Self-dual and complementary dual abelian codes over Galois rings %D 2019 %J Journal of Algebra Combinatorics Discrete Structures and Applications %P -2148-838X %V 6 %N 2 %R doi: 10.13069/jacodesmath.560406 %U 10.13069/jacodesmath.560406
ISNAD Ji̇tman, Somphong , Li̇ng, San . "Self-dual and complementary dual abelian codes over Galois rings". Journal of Algebra Combinatorics Discrete Structures and Applications 6 / 2 (May 2019): 75-94 . https://doi.org/10.13069/jacodesmath.560406
AMA Ji̇tman S , Li̇ng S . Self-dual and complementary dual abelian codes over Galois rings. Journal of Algebra Combinatorics Discrete Structures and Applications. 2019; 6(2): 75-94.
Vancouver Ji̇tman S , Li̇ng S . Self-dual and complementary dual abelian codes over Galois rings. Journal of Algebra Combinatorics Discrete Structures and Applications. 2019; 6(2): 75-94.