On self-dual cyclic codes of length p^a over GR(p^2,s)

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May 2016, 10(2): 255-273. doi: 10.3934/amc.2016004

On self-dual cyclic codes of length $p^a$ over $GR(p^2,s)$

Somphong Jitman ^1, , San Ling ^2, and Ekkasit Sangwisut ^3,

1.	Department of Mathematics, Faculty of Science, Silpakorn University, Nakhon Pathom 73000, Thailand
2.	Division of Mathematical Sciences, School of Physical and Mathematical Sciences, Nanyang Technological University, 21 Nanyang Link, Singapore 637371
3.	Department of Mathematics and Statistics, Faculty of Science, Thaksin University, Phatthalung Campus, Phatthalung 93110, Thailand

Received January 2014 Published April 2016

Abstract
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In this paper, cyclic codes over the Galois ring ${\rm GR}({p^2},s)$ are studied. The main result is the characterization and enumeration of Hermitian self-dual cyclic codes of length $p^a$ over ${\rm GR}({p^2},s)$. Combining with some known results and the standard Discrete Fourier Transform decomposition, we arrive at the characterization and enumeration of Euclidean self-dual cyclic codes of any length over ${\rm GR}({p^2},s)$.

Keywords: codes over rings, Galois rings, Self-dual codes, Hermitian inner product., cyclic codes.

Mathematics Subject Classification: Primary: 94B15, 94B60; Secondary: 13B2.

Citation: Somphong Jitman, San Ling, Ekkasit Sangwisut. On self-dual cyclic codes of length $p^a$ over $GR(p^2,s)$. Advances in Mathematics of Communications, 2016, 10 (2) : 255-273. doi: 10.3934/amc.2016004

References:

[1]	T. Abualrub and R. Oehmke, On the generators of $\mathbb Z_4$ cyclic codes of length $2^e$, IEEE Trans. Inf. Theory, 49 (2003), 2126-2133. doi: 10.1109/TIT.2003.815763.
[2]	A. T. Benjamin and J. J. Quinn, Proofs that Really Count: The Art of Combinatorial Proof, Math. Assoc. Amer., Washington, DC, 2003.
[3]	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.
[4]	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.
[5]	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.
[6]	A. R. Hammons Jr., 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. Inf. Theory, 40 (1994), 301-319. doi: 10.1109/18.312154.
[7]	Y. Jia, S. Ling and C. Xing, On self-dual cyclic codes over finite fields, IEEE Trans. Inf. Theory, 57 (2011), 2243-2251. doi: 10.1109/TIT.2010.2092415.
[8]	S, Jitman, S. Ling, H. Liu and X. Xie, Abelian codes in principal ideal group algebras, IEEE Trans. Inf. Theory, 59 (2013), 3046-3058. doi: 10.1109/TIT.2012.2236383.
[9]	H. M. Kiah, K. H. Leung and S. Ling, Cyclic codes over $GR(p^2,m)$ of length $p^k$, Finite Fields Appl., 14 (2008), 834-846. doi: 10.1016/j.ffa.2008.02.003.
[10]	H. M. Kiah, K. H. Leung and S. Ling, A note on cyclic codes over $GR(p^2,m)$ of length $p^k$, Des. Codes Crypt., 63 (2012), 105-112. doi: 10.1007/s10623-011-9538-5.
[11]	G. Nebe, E. M. Rains and N. J. A. Sloane, Self-Dual Codes and Invariant Theory, Springer-Verlag, Berlin, 2006.
[12]	R. Sobhani and M. Esmaeili, A note on cyclic codes over $GR(p^2,m)$ of length $p^k$, Finite Fields Appl., 15 (2009), 387-391. doi: 10.1016/j.ffa.2009.01.004.
[13]	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.
[14]	Z. X. Wan, Lectures on Finite Fields and Galois Rings, World Scientific, New Jersey, 2003. doi: 10.1142/5350.

show all references

References:

[1]	T. Abualrub and R. Oehmke, On the generators of $\mathbb Z_4$ cyclic codes of length $2^e$, IEEE Trans. Inf. Theory, 49 (2003), 2126-2133. doi: 10.1109/TIT.2003.815763.
[2]	A. T. Benjamin and J. J. Quinn, Proofs that Really Count: The Art of Combinatorial Proof, Math. Assoc. Amer., Washington, DC, 2003.
[3]	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.
[4]	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.
[5]	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.
[6]	A. R. Hammons Jr., 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. Inf. Theory, 40 (1994), 301-319. doi: 10.1109/18.312154.
[7]	Y. Jia, S. Ling and C. Xing, On self-dual cyclic codes over finite fields, IEEE Trans. Inf. Theory, 57 (2011), 2243-2251. doi: 10.1109/TIT.2010.2092415.
[8]	S, Jitman, S. Ling, H. Liu and X. Xie, Abelian codes in principal ideal group algebras, IEEE Trans. Inf. Theory, 59 (2013), 3046-3058. doi: 10.1109/TIT.2012.2236383.
[9]	H. M. Kiah, K. H. Leung and S. Ling, Cyclic codes over $GR(p^2,m)$ of length $p^k$, Finite Fields Appl., 14 (2008), 834-846. doi: 10.1016/j.ffa.2008.02.003.
[10]	H. M. Kiah, K. H. Leung and S. Ling, A note on cyclic codes over $GR(p^2,m)$ of length $p^k$, Des. Codes Crypt., 63 (2012), 105-112. doi: 10.1007/s10623-011-9538-5.
[11]	G. Nebe, E. M. Rains and N. J. A. Sloane, Self-Dual Codes and Invariant Theory, Springer-Verlag, Berlin, 2006.
[12]	R. Sobhani and M. Esmaeili, A note on cyclic codes over $GR(p^2,m)$ of length $p^k$, Finite Fields Appl., 15 (2009), 387-391. doi: 10.1016/j.ffa.2009.01.004.
[13]	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.
[14]	Z. X. Wan, Lectures on Finite Fields and Galois Rings, World Scientific, New Jersey, 2003. doi: 10.1142/5350.

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