February  2012, 6(1): 39-63. doi: 10.3934/amc.2012.6.39

Skew constacyclic codes over finite chain rings

1. 

Department of Mathematics and Computer Science, Faculty of Science, Chulalongkorn University, Bangkok 10330, Thailand, and, Division of Mathematical Sciences, School of Physical and Mathematical Sciences, Nanyang Technological University, 21 Nanyang Link, Singapore 637371, Singapore

2. 

Division of Mathematical Sciences, School of Physical and Mathematical Sciences, Nanyang Technological University, 21 Nanyang Link, Singapore 637371

3. 

Department of Mathematics and Computer Science, Faculty of Science, Chulalongkorn University, Bangkok 10330, Thailand

Received  September 2010 Revised  September 2011 Published  January 2012

Skew polynomial rings over finite fields and over Galois rings have recently been used to study codes. In this paper, we extend this concept to finite chain rings. Properties of skew constacyclic codes generated by monic right divisors of $x^n-\lambda$, where $\lambda$ is a unit element, are exhibited. When $\lambda^2=1$, the generators of Euclidean and Hermitian dual codes of such codes are determined together with necessary and sufficient conditions for them to be Euclidean and Hermitian self-dual. Specializing to codes over the ring $\mathbb F$pm$+u\mathbb F$pm, the structure of all skew constacyclic codes is completely determined. This allows us to express the generators of Euclidean and Hermitian dual codes of skew cyclic and skew negacyclic codes in terms of the generators of the original codes. An illustration of all skew cyclic codes of length $2$ over $\mathbb F_3 + u\mathbb F_3$ and their Euclidean and Hermitian dual codes is also provided.
Citation: 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
References:
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Y. Alkhamees, The group of automorphisms of finite chain rings,, Arab Gulf J. Sci. Res., 8 (1990), 17. Google Scholar

[2]

Y. Alkhamees, The determination of the group of automorphisms of a finite chain ring of characteristic $p$,, Q. J. Math., 42 (1991), 387. doi: 10.1093/qmath/42.1.387. Google Scholar

[3]

M. C. V. Amarra and F. R. Nemenzo, On $(1-u)$-cyclic codes over $\mathbb F$pk $+ u\mathbb F$pk,, Appl. Math. Letters, 21 (2008), 1129. doi: 10.1016/j.aml.2007.07.035. Google Scholar

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C. Bachoc, Application of coding theory to the construction of modular lattices,, J. Combin. Theory Ser. A, 78 (1997), 92. doi: 10.1006/jcta.1996.2763. Google Scholar

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G. Bini and F. Flamini, "Finite Commutative Rings and their Applications,", Kluwer Academic Publishers, (2002). Google Scholar

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A. Bonnecaze and P. Udaya, Cyclic codes and self-dual codes over $\mathbb F_2 + u\mathbb F_2$,, IEEE Trans. Inform. Theory, 45 (1999), 1250. doi: 10.1109/18.761278. Google Scholar

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D. Boucher, W. Geiselmann and F. Ulmer, Skew-cyclic codes,, Appl. Algebra Engin. Commun. Comput., 18 (2007), 379. doi: 10.1007/s00200-007-0043-z. Google Scholar

[8]

D. Boucher, P. Solé and F. Ulmer, Skew constacyclic codes over Galois rings,, Adv. Math. Commun., 2 (2008), 273. doi: 10.3934/amc.2008.2.273. Google Scholar

[9]

D. Boucher and F. Ulmer, Codes as modules over skew polynomial rings,, Lecture Notes Comput. Sci., 5921 (2009), 38. doi: 10.1007/978-3-642-10868-6_3. Google Scholar

[10]

D. Boucher and F. Ulmer, Coding with skew polynomial rings,, J. Symbol. Comput., 44 (2009), 1644. doi: 10.1016/j.jsc.2007.11.008. Google Scholar

[11]

W. E. Clark and D. A. Drake, Finite chain rings,, Abh. Math. Sem. Univ. Hamburg, 39 (1973), 147. doi: 10.1007/BF02992827. Google Scholar

[12]

W. E. Clark and J. J. Liang, Enumeration of finite commutative chain rings,, J. Algebra, 27 (1973), 445. doi: 10.1016/0021-8693(73)90055-0. Google Scholar

[13]

H. Q. Dinh, Negacyclic codes of length $2^s$ over Galois rings,, IEEE Trans. Inform. Theory, 51 (2005), 4252. doi: 10.1109/TIT.2005.859284. Google Scholar

[14]

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. doi: 10.1109/TIT.2009.2013015. Google Scholar

[15]

H. Q. Dinh, Constacyclic codes of length $p^s$ over $\mathbb F$pm $+ u\mathbb F$pm,, J. Algebra, 324 (2010), 940. doi: 10.1016/j.jalgebra.2010.05.027. Google Scholar

[16]

H. Q. Dinh and S. R. López-Permouth, Cyclic and negacyclic codes over finite chain rings,, IEEE Trans. Inform. Theory, 50 (2004), 1728. doi: 10.1109/TIT.2004.831789. Google Scholar

[17]

M. Grassl, Bounds on the minimum distance of linear codes and quantum codes,, available online at \url{http://www.codetables.de} (accessed on 2011-06-01)., (): 2011. Google Scholar

[18]

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. doi: 10.1109/18.312154. Google Scholar

[19]

T. Y. Lam, "Lectures on Modules and Rings,'', Springer-Verlag, (1999). Google Scholar

[20]

B. R. McDonald, "Finite Rings with Identity,", Marcel Dekker, (1974). Google Scholar

[21]

G. H. Norton and A. Sălăgean, On the structure of linear and cyclic codes over a finite chain ring,, Appl. Algebra Engin. Commun. Comput., 10 (2000), 489. doi: 10.1007/PL00012382. Google Scholar

[22]

J. F. Qian, L. N. Zhang and S. X. Zhu, $(1+u)$-cyclic and cyclic codes over the ring $\mathbb F_2 + u\mathbb F_2$,, Appl. Math. Letters, 19 (2006), 820. doi: 10.1016/j.aml.2005.10.011. Google Scholar

[23]

P. Ribenboim, Sur la localisation des anneaux non commutatifs (French),, in, (1972). Google Scholar

[24]

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

[25]

P. Udaya and A. Bonnecaze, Decoding of cyclic codes over $\mathbb F_2 +u\mathbb F_2$,, IEEE Trans. Inform. Theory, 45 (1999), 2148. doi: 10.1109/18.782165. Google Scholar

[26]

P. Udaya and M. U. Siddiqi, Optimal large linear complexity frequency hopping patterns derived from polynomial residue class rings,, IEEE Trans. Inform. Theory, 44 (1998), 1492. doi: 10.1109/18.681324. Google Scholar

[27]

Z.-X. Wan, "Lectures on Finite Fields and Galois Rings,", World Scientific, (2003). Google Scholar

show all references

References:
[1]

Y. Alkhamees, The group of automorphisms of finite chain rings,, Arab Gulf J. Sci. Res., 8 (1990), 17. Google Scholar

[2]

Y. Alkhamees, The determination of the group of automorphisms of a finite chain ring of characteristic $p$,, Q. J. Math., 42 (1991), 387. doi: 10.1093/qmath/42.1.387. Google Scholar

[3]

M. C. V. Amarra and F. R. Nemenzo, On $(1-u)$-cyclic codes over $\mathbb F$pk $+ u\mathbb F$pk,, Appl. Math. Letters, 21 (2008), 1129. doi: 10.1016/j.aml.2007.07.035. Google Scholar

[4]

C. Bachoc, Application of coding theory to the construction of modular lattices,, J. Combin. Theory Ser. A, 78 (1997), 92. doi: 10.1006/jcta.1996.2763. Google Scholar

[5]

G. Bini and F. Flamini, "Finite Commutative Rings and their Applications,", Kluwer Academic Publishers, (2002). Google Scholar

[6]

A. Bonnecaze and P. Udaya, Cyclic codes and self-dual codes over $\mathbb F_2 + u\mathbb F_2$,, IEEE Trans. Inform. Theory, 45 (1999), 1250. doi: 10.1109/18.761278. Google Scholar

[7]

D. Boucher, W. Geiselmann and F. Ulmer, Skew-cyclic codes,, Appl. Algebra Engin. Commun. Comput., 18 (2007), 379. doi: 10.1007/s00200-007-0043-z. Google Scholar

[8]

D. Boucher, P. Solé and F. Ulmer, Skew constacyclic codes over Galois rings,, Adv. Math. Commun., 2 (2008), 273. doi: 10.3934/amc.2008.2.273. Google Scholar

[9]

D. Boucher and F. Ulmer, Codes as modules over skew polynomial rings,, Lecture Notes Comput. Sci., 5921 (2009), 38. doi: 10.1007/978-3-642-10868-6_3. Google Scholar

[10]

D. Boucher and F. Ulmer, Coding with skew polynomial rings,, J. Symbol. Comput., 44 (2009), 1644. doi: 10.1016/j.jsc.2007.11.008. Google Scholar

[11]

W. E. Clark and D. A. Drake, Finite chain rings,, Abh. Math. Sem. Univ. Hamburg, 39 (1973), 147. doi: 10.1007/BF02992827. Google Scholar

[12]

W. E. Clark and J. J. Liang, Enumeration of finite commutative chain rings,, J. Algebra, 27 (1973), 445. doi: 10.1016/0021-8693(73)90055-0. Google Scholar

[13]

H. Q. Dinh, Negacyclic codes of length $2^s$ over Galois rings,, IEEE Trans. Inform. Theory, 51 (2005), 4252. doi: 10.1109/TIT.2005.859284. Google Scholar

[14]

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. doi: 10.1109/TIT.2009.2013015. Google Scholar

[15]

H. Q. Dinh, Constacyclic codes of length $p^s$ over $\mathbb F$pm $+ u\mathbb F$pm,, J. Algebra, 324 (2010), 940. doi: 10.1016/j.jalgebra.2010.05.027. Google Scholar

[16]

H. Q. Dinh and S. R. López-Permouth, Cyclic and negacyclic codes over finite chain rings,, IEEE Trans. Inform. Theory, 50 (2004), 1728. doi: 10.1109/TIT.2004.831789. Google Scholar

[17]

M. Grassl, Bounds on the minimum distance of linear codes and quantum codes,, available online at \url{http://www.codetables.de} (accessed on 2011-06-01)., (): 2011. Google Scholar

[18]

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. doi: 10.1109/18.312154. Google Scholar

[19]

T. Y. Lam, "Lectures on Modules and Rings,'', Springer-Verlag, (1999). Google Scholar

[20]

B. R. McDonald, "Finite Rings with Identity,", Marcel Dekker, (1974). Google Scholar

[21]

G. H. Norton and A. Sălăgean, On the structure of linear and cyclic codes over a finite chain ring,, Appl. Algebra Engin. Commun. Comput., 10 (2000), 489. doi: 10.1007/PL00012382. Google Scholar

[22]

J. F. Qian, L. N. Zhang and S. X. Zhu, $(1+u)$-cyclic and cyclic codes over the ring $\mathbb F_2 + u\mathbb F_2$,, Appl. Math. Letters, 19 (2006), 820. doi: 10.1016/j.aml.2005.10.011. Google Scholar

[23]

P. Ribenboim, Sur la localisation des anneaux non commutatifs (French),, in, (1972). Google Scholar

[24]

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

[25]

P. Udaya and A. Bonnecaze, Decoding of cyclic codes over $\mathbb F_2 +u\mathbb F_2$,, IEEE Trans. Inform. Theory, 45 (1999), 2148. doi: 10.1109/18.782165. Google Scholar

[26]

P. Udaya and M. U. Siddiqi, Optimal large linear complexity frequency hopping patterns derived from polynomial residue class rings,, IEEE Trans. Inform. Theory, 44 (1998), 1492. doi: 10.1109/18.681324. Google Scholar

[27]

Z.-X. Wan, "Lectures on Finite Fields and Galois Rings,", World Scientific, (2003). Google Scholar

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