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:
[1]

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

[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.

[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.

[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.

[5]

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

[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.

[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.

[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.

[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.

[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.

[11]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[19]

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

[20]

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

[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.

[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.

[23]

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

[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.

[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.

[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.

[27]

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

show all references

References:
[1]

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

[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.

[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.

[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.

[5]

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

[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.

[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.

[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.

[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.

[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.

[11]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[19]

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

[20]

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

[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.

[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.

[23]

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

[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.

[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.

[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.

[27]

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

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