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http://dx.doi.org/10.4134/BKMS.b180721

NEGACYCLIC CODES OF LENGTH 8ps OVER Fpm + uFpm  

Klin-eam, Chakkrid (Department of Mathematics Faculty of Science Naresuan University)
Phuto, Jirayu (Department of Mathematics Faculty of Science Naresuan University)
Publication Information
Bulletin of the Korean Mathematical Society / v.56, no.6, 2019 , pp. 1385-1422 More about this Journal
Abstract
Let p be an odd prime. The algebraic structure of all negacyclic codes of length $8_{p^s}$ over the finite commutative chain ring ${\mathbb{F}}_{p^m}+u{\mathbb{F}}_{p^m}$ where $u^2=0$ is studied in this paper. Moreover, we classify all negacyclic codes of length $8_{p^s}$ over ${\mathbb{F}}_{p^m}+u{\mathbb{F}}_{p^m}$ into 5 cases, i.e., $p^m{\equiv}1$ (mod 16), $p^m{\equiv}3$, 11 (mod 16), $p^m{\equiv}5$, 13 (mod 16), $p^m{\equiv}7$, 15 (mod 16) and $p^m{\equiv}9$ (mod 16). From that, the structures of dual and some self-dual negacyclic codes and number of codewords of negacyclic codes are obtained.
Keywords
negacyclic codes; finite chain rings; constacyclic codes; repeated-root codes;
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1 T. Abualrub and I. Siap, Constacyclic codes over $\mathbb{F}_{2}\;+u\mathbb{F}_{2}$, J. Franklin Inst. 346 (2009), no. 5, 520-529. https://doi.org/10.1016/j.jfranklin.2009.02.001   DOI
2 G. Castagnoli, J. L. Massey, P. A. Schoeller, and N. von Seemann, On repeated-root cyclic codes, IEEE Trans. Inform. Theory 37 (1991), no. 2, 337-342. https://doi.org/10.1109/18.75249   DOI
3 B. Chen, H. Q. Dinh, H. Liu, and L. Wang, Constacyclic codes of length $2p^{s}$ over $\mathbb{F}_{p}m\;+u\mathbb{F}_{p}m$, Finite Fields Appl., 37 (2016), 108-130.   DOI
4 H. Q. Dinh, Negacyclic codes of length $2^s$ over Galois rings, IEEE Trans. Inform. Theory 51 (2005), no. 12, 4252-4262. https://doi.org/10.1109/TIT.2005.859284   DOI
5 H. Q. Dinh, Constacyclic codes of length $2^s$ over Galois extension rings of $\mathbb{F}_{2}\;+u\mathbb{F}_{2}$, IEEE Trans. Inform. Theory 55 (2009), no. 4, 1730-1740. https://doi.org/10.1109/TIT.2009.2013015   DOI
6 H. Q. Dinh, Constacyclic codes of length $p^{s}$ over $\mathbb{F}_{p}m\;+u\mathbb{F}_{p}m$, J. Algebra 324 (2010), no. 5, 940-950. https://doi.org/10.1016/j.jalgebra.2010.05.027   DOI
7 H. Q. Dinh, Repeated-root constacyclic codes of length $2p^s$, Finite Fields Appl. 18 (2012), no. 1, 133-143. https://doi.org/10.1016/j.ffa.2011.07.003   DOI
8 H. Q. Dinh, Structure of repeated-root constacyclic codes of length $3p^s$ and their duals, Discrete Math. 313 (2013), no. 9, 983-991. https://doi.org/10.1016/j.disc.2013.01.024   DOI
9 H. Q. Dinh, Y. Fan, H. Liu, X. Liu, and S. Sriboonchitta, On self-dual constacyclic codes of length $p^{s}$ over $\mathbb{F}_{p}m\;+u\mathbb{F}_{p}m$, Discrete Math. 341 (2018), no. 2, 324-335.   DOI
10 H. Q. Dinh, S. Dhompongsa, and S. Sriboonchitta, On constacyclic codes of length $4p^{s}$ over $\mathbb{F}_{p}m\;+u\mathbb{F}_{p}m$, Discrete Math. 340 (2017), no. 4, 832-849. https://doi.org/10.1016/j.disc.2016.11.014   DOI
11 H. Q. Dinh and S. R. Lopez-Permouth, Cyclic and negacyclic codes over nite chain rings, IEEE Trans. Inform. Theory 50 (2004), no. 8, 1728-1744. https://doi.org/10.1109/TIT.2004.831789   DOI
12 H. Q. Dinh, B. T. Nguyen, and S. Sriboonchitta, Negacyclic codes of length $4p^{s}$ over $\mathbb{F}_{p}m\;+u\mathbb{F}_{p}m$ and their duals, Discrete Math. 341 (2018), no. 4, 1055-1071. https://doi.org/10.1016/j.disc.2017.12.019   DOI
13 H. Q. Dinh, B. T. Nguyen, S. Sriboonchitta, and T. M. Vo, On $({\alpha}\;+\;u{\beta})$-constacyclic codes of length $4p^{s}$ over $\mathbb{F}_{p}m\;+u\mathbb{F}_{p}m*$, J. Algebra Appl. 18 (2019), no. 2, 1950023, 16 pp. https://doi.org/10.1142/S0219498819500233
14 H. Q. Dinh, L. Wang, and S. Zhu, Negacyclic codes of length $2p^{s}$ over $\mathbb{F}_{p}m\;+u\mathbb{F}_{p}m$, Finite Fields Appl. 31 (2015), 178-201. https://doi.org/10.1016/j.ffa.2014.09.003   DOI
15 R. M. Roth and G. Seroussi, On cyclic MDS codes of length q over GF(q), IEEE Trans. Inform. Theory 32 (1986), no. 2, 284-285. https://doi.org/10.1109/TIT.1986.1057151   DOI
16 G. Falkner, B. Kowol, W. Heise, and E. Zehendner, On the existence of cyclic optimal codes, Atti Sem. Mat. Fis. Univ. Modena 28 (1979), no. 2, 326-341.
17 K. Guenda and T. A. Gulliver, Construction of cyclic codes over $\mathbb{F}_{2}\;+u\mathbb{F}_{2}$ for DNA computing, Appl. Algebra Engrg. Comm. Comput. 24 (2013), no. 6, 445-459. https://doi.org/10.1007/s00200-013-0188-x   DOI
18 J. H. van Lint, Repeated-root cyclic codes, IEEE Trans. Inform. Theory 37 (1991), no. 2, 343-345. https://doi.org/10.1109/18.75250   DOI
19 X. Liu and X. Xu, Cyclic and negacyclic codes of length $2p^{s}$ over $\mathbb{F}_{p}m\;+u\mathbb{F}_{p}m$, Acta Math. Sci. Ser. B (Engl. Ed.) 34 (2014), no. 3, 829-839. https://doi.org/10.1016/S0252-9602(14)60053-9