American Institute of Mathematical Sciences

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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

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.  doi: 10.1109/TIT.2003.815763.  Google Scholar

[2]

A. T. Benjamin and J. J. Quinn, Proofs that Really Count: The Art of Combinatorial Proof,, Math. Assoc. Amer., (2003).   Google Scholar

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T. Blackford, Cyclic codes over $\mathbb Z_4$ of oddly even length,, Discrete Appl. Math., 128 (2003), 27.  doi: 10.1016/S0166-218X(02)00434-1.  Google Scholar

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S. T. Dougherty and Y. H. Park, On modular cyclic codes,, Finite Fields Appl., 13 (2007), 31.  doi: 10.1016/j.ffa.2005.06.004.  Google Scholar

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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.  doi: 10.1109/18.312154.  Google Scholar

[7]

Y. Jia, S. Ling and C. Xing, On self-dual cyclic codes over finite fields,, IEEE Trans. Inf. Theory, 57 (2011), 2243.  doi: 10.1109/TIT.2010.2092415.  Google Scholar

[8]

S, Jitman, S. Ling, H. Liu and X. Xie, Abelian codes in principal ideal group algebras,, IEEE Trans. Inf. Theory, 59 (2013), 3046.  doi: 10.1109/TIT.2012.2236383.  Google Scholar

[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.  doi: 10.1016/j.ffa.2008.02.003.  Google Scholar

[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.  doi: 10.1007/s10623-011-9538-5.  Google Scholar

[11]

G. Nebe, E. M. Rains and N. J. A. Sloane, Self-Dual Codes and Invariant Theory,, Springer-Verlag, (2006).   Google Scholar

[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.  doi: 10.1016/j.ffa.2009.01.004.  Google Scholar

[13]

R. Sobhani and M. Esmaeili, Cyclic and negacyclic codes over the Galois ring $GR(p^2,m)$,, Discrete Appl. Math., 157 (2009), 2892.  doi: 10.1016/j.dam.2009.03.001.  Google Scholar

[14]

Z. X. Wan, Lectures on Finite Fields and Galois Rings,, World Scientific, (2003).  doi: 10.1142/5350.  Google Scholar

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.  doi: 10.1109/TIT.2003.815763.  Google Scholar

[2]

A. T. Benjamin and J. J. Quinn, Proofs that Really Count: The Art of Combinatorial Proof,, Math. Assoc. Amer., (2003).   Google Scholar

[3]

T. Blackford, Cyclic codes over $\mathbb Z_4$ of oddly even length,, Discrete Appl. Math., 128 (2003), 27.  doi: 10.1016/S0166-218X(02)00434-1.  Google Scholar

[4]

S. T. Dougherty and S. Ling, Cyclic codes over $\mathbb Z_4$ of even length,, Des. Codes Cryptogr., 39 (2006), 127.  doi: 10.1007/s10623-005-2773-x.  Google Scholar

[5]

S. T. Dougherty and Y. H. Park, On modular cyclic codes,, Finite Fields Appl., 13 (2007), 31.  doi: 10.1016/j.ffa.2005.06.004.  Google Scholar

[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.  doi: 10.1109/18.312154.  Google Scholar

[7]

Y. Jia, S. Ling and C. Xing, On self-dual cyclic codes over finite fields,, IEEE Trans. Inf. Theory, 57 (2011), 2243.  doi: 10.1109/TIT.2010.2092415.  Google Scholar

[8]

S, Jitman, S. Ling, H. Liu and X. Xie, Abelian codes in principal ideal group algebras,, IEEE Trans. Inf. Theory, 59 (2013), 3046.  doi: 10.1109/TIT.2012.2236383.  Google Scholar

[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.  doi: 10.1016/j.ffa.2008.02.003.  Google Scholar

[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.  doi: 10.1007/s10623-011-9538-5.  Google Scholar

[11]

G. Nebe, E. M. Rains and N. J. A. Sloane, Self-Dual Codes and Invariant Theory,, Springer-Verlag, (2006).   Google Scholar

[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.  doi: 10.1016/j.ffa.2009.01.004.  Google Scholar

[13]

R. Sobhani and M. Esmaeili, Cyclic and negacyclic codes over the Galois ring $GR(p^2,m)$,, Discrete Appl. Math., 157 (2009), 2892.  doi: 10.1016/j.dam.2009.03.001.  Google Scholar

[14]

Z. X. Wan, Lectures on Finite Fields and Galois Rings,, World Scientific, (2003).  doi: 10.1142/5350.  Google Scholar

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