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The weight distribution of irreducible cyclic codes associated with decomposable generalized Paley graphs
On $ \sigma $-self-orthogonal constacyclic codes over $ \mathbb F_{p^m}+u\mathbb F_{p^m} $
School of Mathematics and Statistics, Central China Normal University, Wuhan, 430079, China |
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 $.
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. |
show all references
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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