    May 2018, 12(2): 231-262. doi: 10.3934/amc.2018016

## Constacyclic codes of length $np^s$ over $\mathbb{F}_{p^m}+u\mathbb{F}_{p^m}$

 1 School of Mathematics and Statistics, Shandong University of Technology, Zibo, Shandong 255091, China 2 Division of Computational Mathematics and Engineering, Institute for Computational Science, Ton Duc Thang University, Ho Chi Minh City, Vietnam 3 Faculty of Mathematics and Statistics, Ton Duc Thang University, Ho Chi Minh City, Vietnam 4 Department of Mathematical Sciences, Kent State University, 4314 Mahoning Avenue, Warren, OH 44483, USA 5 Chern Institute of Mathematics and LPMC, Nankai University, Tianjin 300071, China 6 School of Mathematics and Statistics, Shandong University of Technology, Zibo, Shandong 255091, China 7 Faculty of Economics, Chiang Mai University, Chiang Mai 52000, Thailand

* Corresponding author: Yuan Cao

Received  January 2016 Revised  November 2017 Published  March 2018

Fund Project: This research is supported in part by National Natural Science Foundation of China (Nos. 11671235, 61571243, 11701336, 11471255) and the National Key Basic Research Program of China (Grant 2013CB834204)

Let $\mathbb{F}_{p^m}$ be a finite field of cardinality $p^m$ and $R = \mathbb{F}_{p^m}[u]/\langle u^2\rangle = \mathbb{F}_{p^m}+u\mathbb{F}_{p^m}$ $(u^2 = 0)$, where $p$ is a prime and $m$ is a positive integer. For any $λ∈ \mathbb{F}_{p^m}^{×}$, an explicit representation for all distinct $λ$-constacyclic codes over $R$ of length $np^s$ is given by a canonical form decomposition for each code, where $s$ and $n$ are arbitrary positive integers satisfying ${\rm gcd}(p,n) = 1$. For any such code, using its canonical form decomposition the representation for the dual code of the code is provided. Moreover, representations for all distinct cyclic codes, negacyclic codes and their dual codes of length $np^s$ over $R$ are obtained, and self-duality for these codes are determined. Finally, all distinct self-dual negacyclic codes over $\mathbb{F}_5+u\mathbb{F}_5$ of length $2· 3^t· 5^s$ are listed for any positive integer $t$.

Citation: Yonglin Cao, Yuan Cao, Hai Q. Dinh, Fang-Wei Fu, Jian Gao, Songsak Sriboonchitta. Constacyclic codes of length $np^s$ over $\mathbb{F}_{p^m}+u\mathbb{F}_{p^m}$. Advances in Mathematics of Communications, 2018, 12 (2) : 231-262. doi: 10.3934/amc.2018016
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##### References:
  T. Abualrub and I. Siap, Constacyclic codes over $\mathbb{F}_2+u\mathbb{F}_2$, J. Franklin Inst., 346 (2009), 520-529. doi: 10.1016/j.jfranklin.2009.02.001.  M. C. V. Amerra and F. R. Nemenzo, On $(1-u)$-cyclic codes over $\mathbb{F}_{p^k}+u\mathbb{F}_{p^k}$, Appl. Math. Lett., 21 (2008), 1129-1133. G. K. Bakshi and M. Raka, Self-dual and self-orthogonal negacyclic codes of length $2p^n$ over a finite field, Finite Fields Appl., 19 (2013), 39-54. doi: 10.1016/j.ffa.2012.10.003.  A. Bonnecaze and P. Udaya, Cyclic codes and self-dual codes over $\mathbb{F}_2+u\mathbb{F}_2$, IEEE Trans. Inform. Theory, 45 (1999), 1250-1255. doi: 10.1109/18.761278.  Y. Cao, A class of 1-generator repeated root quasi-cyclic codes, Des. Codes Cryptogr., 72 (2014), 483-496. doi: 10.1007/s10623-012-9777-0.  Y. Cao and Y. Gao, Repeated root cyclic $\mathbb{F}_q$-linear codes over $\mathbb{F}_{q^l}$, Finite Fields Appl., 31 (2015), 202-227. doi: 10.1016/j.ffa.2014.10.003.  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.  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), 1730-1740. doi: 10.1109/TIT.2009.2013015.  H. Q. Dinh, Constacyclic codes of length $p^s$ over $\mathbb{F}_{p^m}+u \mathbb{F}_{p^m}$, J. Algebra, 324 (2010), 940-950. 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. doi: 10.1016/j.ffa.2014.09.003.  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), 832-849. doi: 10.1016/j.disc.2016.11.014.  H. Q. Dinh, A. Sharma, S. Rani and S. Sriboonchitta, Cyclic and negacyclic codes of length $4p^s$ over $\mathbb{F}_{p^m}+u \mathbb{F}_{p^m}$, J. Algebra Appl., (2017), in press.  H. Q. Dinh and S. R. López-Permouth, Cyclic and negacyclic codes over finite chain rings, IEEE Trans. Inform. Theory, 50 (2004), 1728-1744. doi: 10.1109/TIT.2004.831789.  S. T. Dougherty, P. Gaborit, M. Harada and P. Sole, Type Ⅱ codes over $\mathbb{F}_2+u\mathbb{F}_2$, IEEE Trans. Inform. Theory, 45 (1999), 32-45. doi: 10.1109/18.746770.  S. T. Dougherty, J.-L. Kim, H. Kulosman and H. Liu, Self-dual codes over commutative Frobenius rings, Finite Fields Appl., 16 (2010), 14-26. doi: 10.1016/j.ffa.2009.11.004.  T. A. Gulliver and M. Harada, Construction of optimal Type Ⅳ self-dual codes over $\mathbb{F}_2+u\mathbb{F}_2$, IEEE Trans. Inform. Theory, 45 (1999), 2520-2521. doi: 10.1109/18.796394.  W. C. Huffman, On the decompostion of self-dual codes over $\mathbb{F}_2+u\mathbb{F}_2$ with an automorphism of odd prime number, Finite Fields Appl., 13 (2007), 681-712. doi: 10.1016/j.ffa.2006.02.003.  G. H. Norton, A. Sălăgean, On the structure of linear and cyclic codes over finite chain rings, Appl. Algebra in Engrg. Comm. Comput., 10 (2000), 489-506. doi: 10.1007/PL00012382.  G. H. Norton and A. Sălăgean, On the Hamming distance of linear codes over a finite chain rings, IEEE Trans. Inform. Theory, 46 (2000), 1060-1067. doi: 10.1109/18.841186.  J. F. Qian, L. N. Zhang and S. Zhu, $(1+u)$-constacyclic and cyclic codes over $\mathbb{F}_2+u\mathbb{F}_2$, Appl. Math. Lett., 19 (2006), 820-823. doi: 10.1016/j.aml.2005.10.011.  Z. -X. Wan, Lectures on Finite Fields and Galois Rings, World Scientific Pub Co Inc. 2003. Masaaki Harada, Takuji Nishimura. An extremal singly even self-dual code of length 88. Advances in Mathematics of Communications, 2007, 1 (2) : 261-267. doi: 10.3934/amc.2007.1.261  Sihuang Hu, Gabriele Nebe. There is no $[24,12,9]$ doubly-even self-dual code over $\mathbb F_4$. 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