American Institute of Mathematical Sciences

Advanced
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

[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

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

[1]

Minjia Shi, Daitao Huang, Lin Sok, Patrick Solé. Double circulant self-dual and LCD codes over Galois rings. Advances in Mathematics of Communications, 2019, 13 (1) : 171-183. doi: 10.3934/amc.2019011
[2]

Sergio R. López-Permouth, Steve Szabo. On the Hamming weight of repeated root cyclic and negacyclic codes over Galois rings. Advances in Mathematics of Communications, 2009, 3 (4) : 409-420. doi: 10.3934/amc.2009.3.409
[3]

Nabil Bennenni, Kenza Guenda, Sihem Mesnager. DNA cyclic codes over rings. Advances in Mathematics of Communications, 2017, 11 (1) : 83-98. doi: 10.3934/amc.2017004
[4]

Delphine Boucher, Patrick Solé, Felix Ulmer. Skew constacyclic codes over Galois rings. Advances in Mathematics of Communications, 2008, 2 (3) : 273-292. doi: 10.3934/amc.2008.2.273
[5]

Ekkasit Sangwisut, Somphong Jitman, Patanee Udomkavanich. Constacyclic and quasi-twisted Hermitian self-dual codes over finite fields. Advances in Mathematics of Communications, 2017, 11 (3) : 595-613. doi: 10.3934/amc.2017045
[6]

Steven T. Dougherty, Abidin Kaya, Esengül Saltürk. Cyclic codes over local Frobenius rings of order 16. Advances in Mathematics of Communications, 2017, 11 (1) : 99-114. doi: 10.3934/amc.2017005
[7]

David Grant, Mahesh K. Varanasi. The equivalence of space-time codes and codes defined over finite fields and Galois rings. Advances in Mathematics of Communications, 2008, 2 (2) : 131-145. doi: 10.3934/amc.2008.2.131
[8]

Gabriele Nebe, Wolfgang Willems. On self-dual MRD codes. Advances in Mathematics of Communications, 2016, 10 (3) : 633-642. doi: 10.3934/amc.2016031
[9]

Aicha Batoul, Kenza Guenda, T. Aaron Gulliver. Some constacyclic codes over finite chain rings. Advances in Mathematics of Communications, 2016, 10 (4) : 683-694. doi: 10.3934/amc.2016034
[10]

Somphong Jitman, San Ling, Patanee Udomkavanich. Skew constacyclic codes over finite chain rings. Advances in Mathematics of Communications, 2012, 6 (1) : 39-63. doi: 10.3934/amc.2012.6.39
[11]

Kanat Abdukhalikov. On codes over rings invariant under affine groups. Advances in Mathematics of Communications, 2013, 7 (3) : 253-265. doi: 10.3934/amc.2013.7.253
[12]

Eimear Byrne. On the weight distribution of codes over finite rings. Advances in Mathematics of Communications, 2011, 5 (2) : 395-406. doi: 10.3934/amc.2011.5.395
[13]

Steven T. Dougherty, Esengül Saltürk, Steve Szabo. Codes over local rings of order 16 and binary codes. Advances in Mathematics of Communications, 2016, 10 (2) : 379-391. doi: 10.3934/amc.2016012
[14]

Anderson Silva, C. Polcino Milies. Cyclic codes of length $ 2p^n $ over finite chain rings. Advances in Mathematics of Communications, 2019, 0 (0) : 0-0. doi: 10.3934/amc.2020017
[15]

Steven T. Dougherty, Cristina Fernández-Córdoba. Codes over $\mathbb{Z}_{2^k}$, Gray map and self-dual codes. Advances in Mathematics of Communications, 2011, 5 (4) : 571-588. doi: 10.3934/amc.2011.5.571
[16]

Masaaki Harada, Akihiro Munemasa. Classification of self-dual codes of length 36. Advances in Mathematics of Communications, 2012, 6 (2) : 229-235. doi: 10.3934/amc.2012.6.229
[17]

Stefka Bouyuklieva, Anton Malevich, Wolfgang Willems. On the performance of binary extremal self-dual codes. Advances in Mathematics of Communications, 2011, 5 (2) : 267-274. doi: 10.3934/amc.2011.5.267
[18]

Nikolay Yankov, Damyan Anev, Müberra Gürel. Self-dual codes with an automorphism of order 13. Advances in Mathematics of Communications, 2017, 11 (3) : 635-645. doi: 10.3934/amc.2017047
[19]

Steven T. Dougherty, Cristina Fernández-Córdoba, Roger Ten-Valls, Bahattin Yildiz. Quaternary group ring codes: Ranks, kernels and self-dual codes. Advances in Mathematics of Communications, 2019, 0 (0) : 0-0. doi: 10.3934/amc.2020023
[20]

Amita Sahni, Poonam Trama Sehgal. Enumeration of self-dual and self-orthogonal negacyclic codes over finite fields. Advances in Mathematics of Communications, 2015, 9 (4) : 437-447. doi: 10.3934/amc.2015.9.437

2018 Impact Factor: 0.879

Tools

Metrics

  • PDF downloads (8)
  • HTML views (0)
  • Cited by (1)

Other articles
by authors

[Back to Top]

Copyright © 2019 American Institute of Mathematical Sciences