American Institute of Mathematical Sciences

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May  2012, 6(2): 175-191. doi: 10.3934/amc.2012.6.175

On some classes of constacyclic codes over polynomial residue rings

Hai Q. Dinh 1, and  Hien D. T. Nguyen 1,

1. 

Department of Mathematical Sciences, Kent State University, 4314 Mahoning Avenue, Warren, Ohio 44483, USA, and Department of Mathematics, Vinh University, Vinh, Vietnam, Vietnam

Received  April 2011 Revised  July 2011 Published  April 2012

The polynomial residue ring $\mathcal R_a=\frac{\mathbb F_{2^m}[u]}{\langle u^a \rangle}=\mathbb F_{2^m} + u \mathbb F_{2^m}+ \dots + u^{a - 1}\mathbb F_{2^m}$ is a chain ring with residue field $\mathbb F_{2^m}$, that contains precisely $(2^m-1)2^{m(a-1)}$ units, namely, $\alpha_0+u\alpha_1+\dots+u^{a-1}\alpha_{a-1}$, where $\alpha_0,\alpha_1,\dots,\alpha_{a-1} \in \mathbb F_{2^m}$, $\alpha_0 \neq 0$. Two classes of units of $\mathcal R_a$ are considered, namely, $\lambda=1+u\lambda_1+\dots+u^{a-1}\lambda_{a-1}$, where $\lambda_1, \dots, \lambda_{a-1} \in \mathbb F_{2^m}$, $\lambda_1 \neq 0$; and $\Lambda=\Lambda_0+u\Lambda_1+\dots+u^{a-1}\Lambda_{a-1}$, where $\Lambda_0, \Lambda_1, \dots, \Lambda_{a-1} \in \mathbb F_{2^m}$, $\Lambda_0 \neq 0, \Lambda_1 \neq 0$. Among other results, the structure, Hamming and homogeneous distances of $\Lambda$-constacyclic codes of length $2^s$ over $\mathcal R_a$, and the structure of $\lambda$-constacyclic codes of any length over $\mathcal R_a$ are established.
Keywords: polynomial residue rings, dual codes, homogeneous distance., Galois rings, Hamming distance, Constacyclic codes, chain rings.
Mathematics Subject Classification: Primary: 94B15; Secondary: 12Y0.
Citation: Hai Q. Dinh, Hien D. T. Nguyen. On some classes of constacyclic codes over polynomial residue rings. Advances in Mathematics of Communications, 2012, 6 (2) : 175-191. doi: 10.3934/amc.2012.6.175
References:
[1]

T. Abualrub, A. Ghrayeb and R. Oehmke, A mass formula and rank of $\mathbb Z_4$ cyclic codes of length $2^e$,, IEEE Trans. Inform. Theory, 50 (2004), 3306.  doi: 10.1109/TIT.2004.838109.  Google Scholar

[2]

R. Alfaro, S. Bennett, J. Harvey and C. Thornburg, On distances and self-dual codes over $F_q[u]$/$(u^t)$,, Involve, 2 (2009), 177.  doi: 10.2140/involve.2009.2.177.  Google Scholar

[3]

S. D. Berman, Semisimple cyclic and Abelian codes. II (in Russian),, Kibernetika, 3 (1967), 21.  doi: 10.1007/BF01119999.  Google Scholar

[4]

T. Blackford, Negacyclic codes over $\mathbb Z_4$ of even length,, IEEE Trans. Inform. Theory, 49 (2003), 1417.  doi: 10.1109/TIT.2003.811915.  Google Scholar

[5]

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

[6]

A. Bonnecaze and P. Udaya, 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

[7]

A. R. Calderbank, A. R. Hammons, Jr., P. V. Kumar, N. J. A. Sloane and P. Solé, A linear construction for certain Kerdock and Preparata codes,, Bull. AMS, 29 (1993), 218.   Google Scholar

[8]

G. Castagnoli, J. L. Massey, P. A. Schoeller and N. von Seemann, On repeated-root cyclic codes,, IEEE Trans. Inform. Theory, 37 (1991), 337.  doi: 10.1109/18.75249.  Google Scholar

[9]

I. Constaninescu, "Lineare Codes über Restklassenringen ganzer Zahlen und ihre Automorphismen bezüglich einer verallgemeinerten Hamming-Metrik'' (in German),, Ph.D thesis, (1995).   Google Scholar

[10]

I. Constaninescu and W. Heise, A metric for codes over residue class rings of integers,, Problemy Peredachi Informatsii, 33 (1997), 22.   Google Scholar

[11]

I. Constaninescu, W. Heise and T. Honold, Monomial extensions of isometries between codes over $\mathbb Z_m$,, in, (1996), 98.   Google Scholar

[12]

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

[13]

H. Q. Dinh, On the linear ordering of some classes of negacyclic and cyclic codes and their distance distributions,, Finite Fields Appl., 14 (2008), 22.  doi: 10.1016/j.ffa.2007.07.001.  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 $\mathbb F_{p^m}+u\mathbb F_{p^m}$,, 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]

S. Dougherty, P. Gaborit, M. Harada and P. Solé, Type II codes over $\mathbb F_2+u\mathbb F_2$,, IEEE Trans. Inform. Theory, 45 (1999), 32.  doi: 10.1109/18.746770.  Google Scholar

[18]

G. Falkner, B. Kowol, W. Heise and E. Zehendner, On the existence of cyclic optimal codes,, Atti Sem. Mat. Fis. Univ. Modena, 28 (1979), 326.   Google Scholar

[19]

M. Greferath and S. E. Schmidt, Gray Isometries for Finite Chain Rings and a Nonlinear Ternary $(36, 3^{12}, 15)$ Code,, IEEE Trans. Inform. Theory, 45 (1999), 2522.  doi: 10.1109/18.796395.  Google Scholar

[20]

M. Greferath and S. E. Schmidt, Finite ring combinatorics and MacWilliams's equivalence theorem,, J. Combin. Theory Ser. A, 92 (2000), 17.  doi: 10.1006/jcta.1999.3033.  Google Scholar

[21]

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. Inform. Theory, 40 (1994), 301.  doi: 10.1109/18.312154.  Google Scholar

[22]

W. Heise, T. Honold and A. A. Nechaev, Weighted modules and representations of codes,, in, (1998), 123.   Google Scholar

[23]

T. Honold and I. Landjev, Linear representable codes over chain rings,, in, (1998), 135.   Google Scholar

[24]

W. C. Huffman and V. Pless, "Fundamentals of Error-Correcting Codes,'', Cambridge University Press, (2003).   Google Scholar

[25]

S. Ling and P. Solé, Duadic codes over $\mathbb F_2+u\mathbb F_2$,, Appl. Algebra Engrg. Comm. Comput., 12 (2001), 365.  doi: 10.1007/s002000100079.  Google Scholar

[26]

F. J. MacWilliams, Error-correcting codes for multiple-level transmissions,, Bell System Tech. J., 40 (1961), 281.   Google Scholar

[27]

F. J. MacWilliams, Combinatorial problems of elementary abelian groups,, Ph.D thesis, (1962).   Google Scholar

[28]

J. L. Massey, D. J. Costello and J. Justesen, Polynomial weights and code constructions,, IEEE Trans. Inform. Theory, 19 (1973), 101.  doi: 10.1109/TIT.1973.1054936.  Google Scholar

[29]

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

[30]

A. A. Nechaev, Kerdock code in a cyclic form (in Russian),, Diskr. Math. (USSR), 1 (1989), 123.  doi: 10.1515/dma.1991.1.4.365.  Google Scholar

[31]

C.-S. Nedeloaia, Weight distributions of cyclic self-dual codes,, IEEE Trans. Inform. Theory, 49 (2003), 1582.  doi: 10.1109/TIT.2003.811921.  Google Scholar

[32]

G. Norton and A. Sălăgean-Mandache, On the structure of linear cyclic codes over finite chain rings,, Appl. Algebra Engrg. Comm. Comput., 10 (2000), 489.  doi: 10.1007/PL00012382.  Google Scholar

[33]

M. Ozen and I. Siap, Linear codes over $\mathbb F_q[u]$/$(u^s)$ with respect to the osenbloom-Tsffasman metric,, Des. Codes Cryptogr., 38 (2006), 17.  doi: 10.1007/s10623-004-5658-5.  Google Scholar

[34]

V. Pless and W. C. Huffman, "Handbook of Coding Theory,'', Elsevier, (1998).   Google Scholar

[35]

R. M. Roth and G. Seroussi, On cyclic MDS codes of length $q$ over $\GF(q)$,, IEEE Trans. Inform. Theory, 32 (1986), 284.  doi: 10.1109/TIT.1986.1057151.  Google Scholar

[36]

A. Sălăgean, Repeated-root cyclic and negacyclic codes over finite chain rings,, Discrete Appl. Math., 154 (2006), 413.  doi: 10.1016/j.dam.2005.03.016.  Google Scholar

[37]

L.-Z. Tang, C. B. Soh and E. Gunawan, A note on the $q$-ary image of a $q^m$-ary repeated-root cyclic code,, IEEE Trans. Inform. Theory, 43 (1997), 732.  doi: 10.1109/18.556131.  Google Scholar

[38]

J. H. van Lint, Repeated-root cyclic codes,, IEEE Trans. Inform. Theory, 37 (1991), 343.  doi: 10.1109/18.75250.  Google Scholar

[39]

J. A. Wood, Duality for modules over finite rings and applications to coding theory,, American J. Math., 121 (1999), 555.  doi: 10.1353/ajm.1999.0024.  Google Scholar

[40]

K.-H. Zimmermann, On generalizations of repeated-root cyclic codes,, IEEE Trans. Inform. Theory, 42 (1996), 641.  doi: 10.1109/18.485736.  Google Scholar

show all references

References:
[1]

T. Abualrub, A. Ghrayeb and R. Oehmke, A mass formula and rank of $\mathbb Z_4$ cyclic codes of length $2^e$,, IEEE Trans. Inform. Theory, 50 (2004), 3306.  doi: 10.1109/TIT.2004.838109.  Google Scholar

[2]

R. Alfaro, S. Bennett, J. Harvey and C. Thornburg, On distances and self-dual codes over $F_q[u]$/$(u^t)$,, Involve, 2 (2009), 177.  doi: 10.2140/involve.2009.2.177.  Google Scholar

[3]

S. D. Berman, Semisimple cyclic and Abelian codes. II (in Russian),, Kibernetika, 3 (1967), 21.  doi: 10.1007/BF01119999.  Google Scholar

[4]

T. Blackford, Negacyclic codes over $\mathbb Z_4$ of even length,, IEEE Trans. Inform. Theory, 49 (2003), 1417.  doi: 10.1109/TIT.2003.811915.  Google Scholar

[5]

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

[6]

A. Bonnecaze and P. Udaya, 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

[7]

A. R. Calderbank, A. R. Hammons, Jr., P. V. Kumar, N. J. A. Sloane and P. Solé, A linear construction for certain Kerdock and Preparata codes,, Bull. AMS, 29 (1993), 218.   Google Scholar

[8]

G. Castagnoli, J. L. Massey, P. A. Schoeller and N. von Seemann, On repeated-root cyclic codes,, IEEE Trans. Inform. Theory, 37 (1991), 337.  doi: 10.1109/18.75249.  Google Scholar

[9]

I. Constaninescu, "Lineare Codes über Restklassenringen ganzer Zahlen und ihre Automorphismen bezüglich einer verallgemeinerten Hamming-Metrik'' (in German),, Ph.D thesis, (1995).   Google Scholar

[10]

I. Constaninescu and W. Heise, A metric for codes over residue class rings of integers,, Problemy Peredachi Informatsii, 33 (1997), 22.   Google Scholar

[11]

I. Constaninescu, W. Heise and T. Honold, Monomial extensions of isometries between codes over $\mathbb Z_m$,, in, (1996), 98.   Google Scholar

[12]

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

[13]

H. Q. Dinh, On the linear ordering of some classes of negacyclic and cyclic codes and their distance distributions,, Finite Fields Appl., 14 (2008), 22.  doi: 10.1016/j.ffa.2007.07.001.  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 $\mathbb F_{p^m}+u\mathbb F_{p^m}$,, 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]

S. Dougherty, P. Gaborit, M. Harada and P. Solé, Type II codes over $\mathbb F_2+u\mathbb F_2$,, IEEE Trans. Inform. Theory, 45 (1999), 32.  doi: 10.1109/18.746770.  Google Scholar

[18]

G. Falkner, B. Kowol, W. Heise and E. Zehendner, On the existence of cyclic optimal codes,, Atti Sem. Mat. Fis. Univ. Modena, 28 (1979), 326.   Google Scholar

[19]

M. Greferath and S. E. Schmidt, Gray Isometries for Finite Chain Rings and a Nonlinear Ternary $(36, 3^{12}, 15)$ Code,, IEEE Trans. Inform. Theory, 45 (1999), 2522.  doi: 10.1109/18.796395.  Google Scholar

[20]

M. Greferath and S. E. Schmidt, Finite ring combinatorics and MacWilliams's equivalence theorem,, J. Combin. Theory Ser. A, 92 (2000), 17.  doi: 10.1006/jcta.1999.3033.  Google Scholar

[21]

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. Inform. Theory, 40 (1994), 301.  doi: 10.1109/18.312154.  Google Scholar

[22]

W. Heise, T. Honold and A. A. Nechaev, Weighted modules and representations of codes,, in, (1998), 123.   Google Scholar

[23]

T. Honold and I. Landjev, Linear representable codes over chain rings,, in, (1998), 135.   Google Scholar

[24]

W. C. Huffman and V. Pless, "Fundamentals of Error-Correcting Codes,'', Cambridge University Press, (2003).   Google Scholar

[25]

S. Ling and P. Solé, Duadic codes over $\mathbb F_2+u\mathbb F_2$,, Appl. Algebra Engrg. Comm. Comput., 12 (2001), 365.  doi: 10.1007/s002000100079.  Google Scholar

[26]

F. J. MacWilliams, Error-correcting codes for multiple-level transmissions,, Bell System Tech. J., 40 (1961), 281.   Google Scholar

[27]

F. J. MacWilliams, Combinatorial problems of elementary abelian groups,, Ph.D thesis, (1962).   Google Scholar

[28]

J. L. Massey, D. J. Costello and J. Justesen, Polynomial weights and code constructions,, IEEE Trans. Inform. Theory, 19 (1973), 101.  doi: 10.1109/TIT.1973.1054936.  Google Scholar

[29]

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

[30]

A. A. Nechaev, Kerdock code in a cyclic form (in Russian),, Diskr. Math. (USSR), 1 (1989), 123.  doi: 10.1515/dma.1991.1.4.365.  Google Scholar

[31]

C.-S. Nedeloaia, Weight distributions of cyclic self-dual codes,, IEEE Trans. Inform. Theory, 49 (2003), 1582.  doi: 10.1109/TIT.2003.811921.  Google Scholar

[32]

G. Norton and A. Sălăgean-Mandache, On the structure of linear cyclic codes over finite chain rings,, Appl. Algebra Engrg. Comm. Comput., 10 (2000), 489.  doi: 10.1007/PL00012382.  Google Scholar

[33]

M. Ozen and I. Siap, Linear codes over $\mathbb F_q[u]$/$(u^s)$ with respect to the osenbloom-Tsffasman metric,, Des. Codes Cryptogr., 38 (2006), 17.  doi: 10.1007/s10623-004-5658-5.  Google Scholar

[34]

V. Pless and W. C. Huffman, "Handbook of Coding Theory,'', Elsevier, (1998).   Google Scholar

[35]

R. M. Roth and G. Seroussi, On cyclic MDS codes of length $q$ over $\GF(q)$,, IEEE Trans. Inform. Theory, 32 (1986), 284.  doi: 10.1109/TIT.1986.1057151.  Google Scholar

[36]

A. Sălăgean, Repeated-root cyclic and negacyclic codes over finite chain rings,, Discrete Appl. Math., 154 (2006), 413.  doi: 10.1016/j.dam.2005.03.016.  Google Scholar

[37]

L.-Z. Tang, C. B. Soh and E. Gunawan, A note on the $q$-ary image of a $q^m$-ary repeated-root cyclic code,, IEEE Trans. Inform. Theory, 43 (1997), 732.  doi: 10.1109/18.556131.  Google Scholar

[38]

J. H. van Lint, Repeated-root cyclic codes,, IEEE Trans. Inform. Theory, 37 (1991), 343.  doi: 10.1109/18.75250.  Google Scholar

[39]

J. A. Wood, Duality for modules over finite rings and applications to coding theory,, American J. Math., 121 (1999), 555.  doi: 10.1353/ajm.1999.0024.  Google Scholar

[40]

K.-H. Zimmermann, On generalizations of repeated-root cyclic codes,, IEEE Trans. Inform. Theory, 42 (1996), 641.  doi: 10.1109/18.485736.  Google Scholar

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