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On the ideal associated to a linear code
On self-dual cyclic codes of length $p^a$ over $GR(p^2,s)$
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 |
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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