On \sigma -self-orthogonal constacyclic codes over \mathbb F_{p^m}+u\mathbb F_{p^m}

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August 2022, 16(3): 643-665. doi: 10.3934/amc.2020127

On $ \sigma $-self-orthogonal constacyclic codes over $ \mathbb F_{p^m}+u\mathbb F_{p^m} $

Hongwei Liu and Jingge Liu ^,

School of Mathematics and Statistics, Central China Normal University, Wuhan, 430079, China

^* Corresponding author: Jingge Liu

Received September 2019 Revised March 2020 Published August 2022 Early access December 2020

Fund Project: This work is supported by the National Natural Science Foundation of China (NSFC) under Grant 11871025

In this paper, we generalize the notion of self-orthogonal codes to $ \sigma $-self-orthogonal codes over an arbitrary finite ring. Then, we study the $ \sigma $-self-orthogonality of constacyclic codes of length $ p^s $ over the finite commutative chain ring $ \mathbb F_{p^m} + u \mathbb F_{p^m} $, where $ p $ is a prime, $ u^2 = 0 $ and $ \sigma $ is an arbitrary ring automorphism of $ \mathbb F_{p^m} + u \mathbb F_{p^m} $. We characterize the structure of $ \sigma $-dual code of a $ \lambda $-constacyclic code of length $ p^s $ over $ \mathbb F_{p^m} + u \mathbb F_{p^m} $. Further, the necessary and sufficient conditions for a $ \lambda $-constacyclic code to be $ \sigma $-self-orthogonal are provided. In particular, we determine all $ \sigma $-self-dual constacyclic codes of length $ p^s $ over $ \mathbb F_{p^m} + u \mathbb F_{p^m} $. In the end of this paper, when $ p $ is an odd prime, we extend the results to constacyclic codes of length $ 2 p^s $.

Keywords: Constacyclic code, repeated-root code, $ \sigma $-self-orthogonal code, $ \sigma $-self-dual code, finite commutative chain ring.

Mathematics Subject Classification: Primary: 94A55, 94B05; Secondary: 94B15.

Citation: Hongwei Liu, Jingge Liu. On $ \sigma $-self-orthogonal constacyclic codes over $ \mathbb F_{p^m}+u\mathbb F_{p^m} $. Advances in Mathematics of Communications, 2022, 16 (3) : 643-665. doi: 10.3934/amc.2020127

References:

[1]	E. F. J. Assmus and H. F. J. Mattson, New $5$-designs, J. Combinatorial Theory, 6 (1969), 122-151. doi: 10.1016/S0021-9800(69)80115-8.
[2]	Y. Alkhamees, The determination of the group of automorphisms of a finite chain ring of characteristic $p$, Quart. J. Math. Oxford Ser., 42 (1991), 387-391. doi: 10.1093/qmath/42.1.387.
[3]	C. Bachoc, Applications of coding theory to the construction of modular lattices, J. Combin. Theory Ser. A, 78 (1997), 92-119. doi: 10.1006/jcta.1996.2763.
[4]	E. R. Berlekamp, Algebraic Coding Theory, Mc Graw-Hill Book Company, 1968. doi: 10.1007/0-387-27105-8_9.
[5]	E. R. Berlekamp, Negacyclic codes for the Lee metric, in Proc. Conf. Combinatorial Mathematics and its Applications, Chapel Hill, NC, 1967,298–316. doi: 10.1142/9789814635905_0009.
[6]	T. Blackford, Cyclic codes over $\mathbb{Z}_4$ of oddly even length, Discrete Appl. Math., 128 (2003), 27-46. doi: 10.1016/S0166-218X(02)00434-1.
[7]	A. R. Calderbank, A. R. Hammons, P. V. Kumar, N. J. A. Sloane and P. Solé, A linear construction for certain Kerdock and Preparata codes, Bull. Amer. Math. Soc. (N.S.), 29 (1993), 218-222. doi: 10.1090/S0273-0979-1993-00426-9.
[8]	A. R. Calderbank, E. M. Rains, P. W. Shor and N. J. A. Sloane, Quantum error correction via codes over $GF(4)$, IEEE Trans. Inform. Theory, 44 (1998), 1369-1387. doi: 10.1109/18.681315.
[9]	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: 10.1016/j.ffa.2015.09.006.
[10]	B. Chen, S. Ling and G. Zhang, Application of constacyclic codes to quantum MDS Codes, IEEE Trans. Inform. Theory, 61 (2015), 1474-1484. doi: 10.1109/TIT.2015.2388576.
[11]	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.
[12]	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. doi: 10.1016/j.jalgebra.2010.05.027.
[13]	H. Q. Dinh, Negacyclic codes of length $2^s$ over Galois rings, IEEE Trans. Inform. Theory, 51 (2005), 4252-4262. doi: 10.1109/TIT.2005.859284.
[14]	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), 324-335. doi: 10.1016/j.disc.2017.08.044.
[15]	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.
[16]	S. T. Dougherty and S. Ling, Cyclic codes over $\mathbb{Z}_4$ of even length, Des. Codes Cryptogr., 39 (2006), 127-153. doi: 10.1007/s10623-005-2773-x.
[17]	S. T. Dougherty and Y. H. Park, On modular cyclic codes, Finite Fields Appl., 13 (2007), 31-57. doi: 10.1016/j.ffa.2005.06.004.
[18]	Y. Fan and L. Zhang, Galois self-dual constacyclic codes, Des. Codes Cryptogr., 84 (2017), 473-492. doi: 10.1007/s10623-016-0282-8.
[19]	A. R. Hammons, P. V. Kumar, A. R. Calderbank, N. J. A. Sloane and P. Solé, The $\mathbb{Z}_4$-linearity of Kerdock, Preparata, Goethals, and related codes, IEEE Trans. Inform. Theory, 40 (1994), 301-319. doi: 10.1109/18.312154.
[20]	X. Kai, S. Zhu and P. Li, Constacyclic codes and some new quantum MDS Codes, IEEE Trans. Inform. Theory, 60 (2014), 2080-2086. doi: 10.1109/TIT.2014.2308180.
[21]	H. Liu and Y. Maouche, Some repeated-root constacyclic codes over Galois rings, IEEE Trans. Inform. Theory, 63 (2017), 6247-6255. doi: 10.1109/TIT.2017.2738627.
[22]	H. Liu and X. Pan, Galois hulls of linear codes over finite fields, Des. Codes Cryptogr., 88 (2020), 241-255. doi: 10.1007/s10623-019-00681-2.
[23]	X. Liu, Y. Fan and H. Liu, Galois LCD codes over finite fields, Finite Fields Appl., 49 (2018), 227-242. doi: 10.1016/j.ffa.2017.10.001.
[24]	X. Liu, L. Yu and P. Hu, New entanglement-assisted quantum codes from $k$-Galois dual codes, Finite Fields Appl., 55 (2019), 21-32. doi: 10.1016/j.ffa.2018.09.001.
[25]	A. A. Nechaev, Kerdock code in cyclic form, Discrete Math. Appl., 1 (1991), 365-384. doi: 10.1515/dma.1991.1.4.365.
[26]	V. Pless, A classification of self-orthogonal codes over $\mathbb{F}_2$, Discrete Math., 3 (1972), 209-246. doi: 10.1016/0012-365X(72)90034-9.
[27]	R. Sobhani and M. Esmaeili, Cyclic and negacyclic codes over the Galois ring $GR(p^2, m)$, Discrete Appl. Math., 157 (2009), 2892-2903. doi: 10.1016/j.dam.2009.03.001.

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References:

[1]	E. F. J. Assmus and H. F. J. Mattson, New $5$-designs, J. Combinatorial Theory, 6 (1969), 122-151. doi: 10.1016/S0021-9800(69)80115-8.
[2]	Y. Alkhamees, The determination of the group of automorphisms of a finite chain ring of characteristic $p$, Quart. J. Math. Oxford Ser., 42 (1991), 387-391. doi: 10.1093/qmath/42.1.387.
[3]	C. Bachoc, Applications of coding theory to the construction of modular lattices, J. Combin. Theory Ser. A, 78 (1997), 92-119. doi: 10.1006/jcta.1996.2763.
[4]	E. R. Berlekamp, Algebraic Coding Theory, Mc Graw-Hill Book Company, 1968. doi: 10.1007/0-387-27105-8_9.
[5]	E. R. Berlekamp, Negacyclic codes for the Lee metric, in Proc. Conf. Combinatorial Mathematics and its Applications, Chapel Hill, NC, 1967,298–316. doi: 10.1142/9789814635905_0009.
[6]	T. Blackford, Cyclic codes over $\mathbb{Z}_4$ of oddly even length, Discrete Appl. Math., 128 (2003), 27-46. doi: 10.1016/S0166-218X(02)00434-1.
[7]	A. R. Calderbank, A. R. Hammons, P. V. Kumar, N. J. A. Sloane and P. Solé, A linear construction for certain Kerdock and Preparata codes, Bull. Amer. Math. Soc. (N.S.), 29 (1993), 218-222. doi: 10.1090/S0273-0979-1993-00426-9.
[8]	A. R. Calderbank, E. M. Rains, P. W. Shor and N. J. A. Sloane, Quantum error correction via codes over $GF(4)$, IEEE Trans. Inform. Theory, 44 (1998), 1369-1387. doi: 10.1109/18.681315.
[9]	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: 10.1016/j.ffa.2015.09.006.
[10]	B. Chen, S. Ling and G. Zhang, Application of constacyclic codes to quantum MDS Codes, IEEE Trans. Inform. Theory, 61 (2015), 1474-1484. doi: 10.1109/TIT.2015.2388576.
[11]	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.
[12]	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. doi: 10.1016/j.jalgebra.2010.05.027.
[13]	H. Q. Dinh, Negacyclic codes of length $2^s$ over Galois rings, IEEE Trans. Inform. Theory, 51 (2005), 4252-4262. doi: 10.1109/TIT.2005.859284.
[14]	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), 324-335. doi: 10.1016/j.disc.2017.08.044.
[15]	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.
[16]	S. T. Dougherty and S. Ling, Cyclic codes over $\mathbb{Z}_4$ of even length, Des. Codes Cryptogr., 39 (2006), 127-153. doi: 10.1007/s10623-005-2773-x.
[17]	S. T. Dougherty and Y. H. Park, On modular cyclic codes, Finite Fields Appl., 13 (2007), 31-57. doi: 10.1016/j.ffa.2005.06.004.
[18]	Y. Fan and L. Zhang, Galois self-dual constacyclic codes, Des. Codes Cryptogr., 84 (2017), 473-492. doi: 10.1007/s10623-016-0282-8.
[19]	A. R. Hammons, P. V. Kumar, A. R. Calderbank, N. J. A. Sloane and P. Solé, The $\mathbb{Z}_4$-linearity of Kerdock, Preparata, Goethals, and related codes, IEEE Trans. Inform. Theory, 40 (1994), 301-319. doi: 10.1109/18.312154.
[20]	X. Kai, S. Zhu and P. Li, Constacyclic codes and some new quantum MDS Codes, IEEE Trans. Inform. Theory, 60 (2014), 2080-2086. doi: 10.1109/TIT.2014.2308180.
[21]	H. Liu and Y. Maouche, Some repeated-root constacyclic codes over Galois rings, IEEE Trans. Inform. Theory, 63 (2017), 6247-6255. doi: 10.1109/TIT.2017.2738627.
[22]	H. Liu and X. Pan, Galois hulls of linear codes over finite fields, Des. Codes Cryptogr., 88 (2020), 241-255. doi: 10.1007/s10623-019-00681-2.
[23]	X. Liu, Y. Fan and H. Liu, Galois LCD codes over finite fields, Finite Fields Appl., 49 (2018), 227-242. doi: 10.1016/j.ffa.2017.10.001.
[24]	X. Liu, L. Yu and P. Hu, New entanglement-assisted quantum codes from $k$-Galois dual codes, Finite Fields Appl., 55 (2019), 21-32. doi: 10.1016/j.ffa.2018.09.001.
[25]	A. A. Nechaev, Kerdock code in cyclic form, Discrete Math. Appl., 1 (1991), 365-384. doi: 10.1515/dma.1991.1.4.365.
[26]	V. Pless, A classification of self-orthogonal codes over $\mathbb{F}_2$, Discrete Math., 3 (1972), 209-246. doi: 10.1016/0012-365X(72)90034-9.
[27]	R. Sobhani and M. Esmaeili, Cyclic and negacyclic codes over the Galois ring $GR(p^2, m)$, Discrete Appl. Math., 157 (2009), 2892-2903. doi: 10.1016/j.dam.2009.03.001.

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