# American Institute of Mathematical Sciences

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

## On some classes of constacyclic codes over polynomial residue rings

 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.
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), 3306-3312. doi: 10.1109/TIT.2004.838109. [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), 177-194. doi: 10.2140/involve.2009.2.177. [3] S. D. Berman, Semisimple cyclic and Abelian codes. II (in Russian), Kibernetika, 3 (1967), 21-30; English translation: Cybernetics, 3 (1967), 17-23. doi: 10.1007/BF01119999. [4] T. Blackford, Negacyclic codes over $\mathbb Z_4$ of even length, IEEE Trans. Inform. Theory, 49 (2003), 1417-1424. doi: 10.1109/TIT.2003.811915. [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), 1250-1255. doi: 10.1109/18.761278. [6] A. Bonnecaze and P. Udaya, Decoding of cyclic codes over $\mathbb F_2+u\mathbb F_2$, IEEE Trans. Inform. Theory, 45 (1999), 2148-2157. doi: 10.1109/18.782165. [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), 218-222. [8] G. Castagnoli, J. L. Massey, P. A. Schoeller and N. von Seemann, On repeated-root cyclic codes, IEEE Trans. Inform. Theory, 37 (1991), 337-342. doi: 10.1109/18.75249. [9] I. Constaninescu, "Lineare Codes über Restklassenringen ganzer Zahlen und ihre Automorphismen bezüglich einer verallgemeinerten Hamming-Metrik'' (in German), Ph.D thesis, Technische Universität, München, Germany, 1995. [10] I. Constaninescu and W. Heise, A metric for codes over residue class rings of integers, Problemy Peredachi Informatsii, 33 (1997), 22-28. [11] I. Constaninescu, W. Heise and T. Honold, Monomial extensions of isometries between codes over $\mathbb Z_m$, in "Proceedings of the 5th International Workshop on Algebraic and Combinatorial Coding Theory (ACCT'96),'' Unicorn Shumen, (1996), 98-104. [12] H. Q. Dinh, Negacyclic codes of length $2^s$ over Galois rings, IEEE Trans. Inform. Theory, 51 (2005), 4252-4262. doi: 10.1109/TIT.2005.859284. [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), 22-40. doi: 10.1016/j.ffa.2007.07.001. [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), 1730-1740. doi: 10.1109/TIT.2009.2013015. [15] H. Q. Dinh, Constacyclic codes of length $p^s$ over $\mathbb F_{p^m}+u\mathbb F_{p^m}$, J. Algebra, 324 (2010), 940-950. 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), 1728-1744. doi: 10.1109/TIT.2004.831789. [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), 32-45. doi: 10.1109/18.746770. [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), 326-341. [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), 2522-2524. doi: 10.1109/18.796395. [20] M. Greferath and S. E. Schmidt, Finite ring combinatorics and MacWilliams's equivalence theorem, J. Combin. Theory Ser. A, 92 (2000), 17-28. doi: 10.1006/jcta.1999.3033. [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), 301-319. doi: 10.1109/18.312154. [22] W. Heise, T. Honold and A. A. Nechaev, Weighted modules and representations of codes, in "Proceedings of the ACCT 6,'' Pskov, Russia, (1998), 123-129. [23] T. Honold and I. Landjev, Linear representable codes over chain rings, in "Proceedings of the ACCT 6,'' Pskov, Russia, (1998), 135-141. [24] W. C. Huffman and V. Pless, "Fundamentals of Error-Correcting Codes,'' Cambridge University Press, Cambridge, 2003. [25] S. Ling and P. Solé, Duadic codes over $\mathbb F_2+u\mathbb F_2$, Appl. Algebra Engrg. Comm. Comput., 12 (2001), 365-379. doi: 10.1007/s002000100079. [26] F. J. MacWilliams, Error-correcting codes for multiple-level transmissions, Bell System Tech. J., 40 (1961), 281-308. [27] F. J. MacWilliams, Combinatorial problems of elementary abelian groups, Ph.D thesis, Harvard University, Cambridge, MA, 1962. [28] J. L. Massey, D. J. Costello and J. Justesen, Polynomial weights and code constructions, IEEE Trans. Inform. Theory, 19 (1973), 101-110. doi: 10.1109/TIT.1973.1054936. [29] B. R. McDonald, "Finite Rings with Identity,'' Marcel Dekker, New York, 1974. [30] A. A. Nechaev, Kerdock code in a cyclic form (in Russian), Diskr. Math. (USSR), 1 (1989), 123-139; English translation: Discrete Math. Appl., 1 (1991), 365-384. doi: 10.1515/dma.1991.1.4.365. [31] C.-S. Nedeloaia, Weight distributions of cyclic self-dual codes, IEEE Trans. Inform. Theory, 49 (2003), 1582-1591. doi: 10.1109/TIT.2003.811921. [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), 489-506. doi: 10.1007/PL00012382. [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), 17-29. doi: 10.1007/s10623-004-5658-5. [34] V. Pless and W. C. Huffman, "Handbook of Coding Theory,'' Elsevier, Amsterdam, 1998. [35] R. M. Roth and G. Seroussi, On cyclic MDS codes of length $q$ over $\GF(q)$, IEEE Trans. Inform. Theory, 32 (1986), 284-285. doi: 10.1109/TIT.1986.1057151. [36] A. Sălăgean, Repeated-root cyclic and negacyclic codes over finite chain rings, Discrete Appl. Math., 154 (2006), 413-419. doi: 10.1016/j.dam.2005.03.016. [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), 732-737. doi: 10.1109/18.556131. [38] J. H. van Lint, Repeated-root cyclic codes, IEEE Trans. Inform. Theory, 37 (1991), 343-345. doi: 10.1109/18.75250. [39] J. A. Wood, Duality for modules over finite rings and applications to coding theory, American J. Math., 121 (1999), 555-575. doi: 10.1353/ajm.1999.0024. [40] K.-H. Zimmermann, On generalizations of repeated-root cyclic codes, IEEE Trans. Inform. Theory, 42 (1996), 641-649. doi: 10.1109/18.485736.

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##### 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), 3306-3312. doi: 10.1109/TIT.2004.838109. [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), 177-194. doi: 10.2140/involve.2009.2.177. [3] S. D. Berman, Semisimple cyclic and Abelian codes. II (in Russian), Kibernetika, 3 (1967), 21-30; English translation: Cybernetics, 3 (1967), 17-23. doi: 10.1007/BF01119999. [4] T. Blackford, Negacyclic codes over $\mathbb Z_4$ of even length, IEEE Trans. Inform. Theory, 49 (2003), 1417-1424. doi: 10.1109/TIT.2003.811915. [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), 1250-1255. doi: 10.1109/18.761278. [6] A. Bonnecaze and P. Udaya, Decoding of cyclic codes over $\mathbb F_2+u\mathbb F_2$, IEEE Trans. Inform. Theory, 45 (1999), 2148-2157. doi: 10.1109/18.782165. [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), 218-222. [8] G. Castagnoli, J. L. Massey, P. A. Schoeller and N. von Seemann, On repeated-root cyclic codes, IEEE Trans. Inform. Theory, 37 (1991), 337-342. doi: 10.1109/18.75249. [9] I. Constaninescu, "Lineare Codes über Restklassenringen ganzer Zahlen und ihre Automorphismen bezüglich einer verallgemeinerten Hamming-Metrik'' (in German), Ph.D thesis, Technische Universität, München, Germany, 1995. [10] I. Constaninescu and W. Heise, A metric for codes over residue class rings of integers, Problemy Peredachi Informatsii, 33 (1997), 22-28. [11] I. Constaninescu, W. Heise and T. Honold, Monomial extensions of isometries between codes over $\mathbb Z_m$, in "Proceedings of the 5th International Workshop on Algebraic and Combinatorial Coding Theory (ACCT'96),'' Unicorn Shumen, (1996), 98-104. [12] H. Q. Dinh, Negacyclic codes of length $2^s$ over Galois rings, IEEE Trans. Inform. Theory, 51 (2005), 4252-4262. doi: 10.1109/TIT.2005.859284. [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), 22-40. doi: 10.1016/j.ffa.2007.07.001. [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), 1730-1740. doi: 10.1109/TIT.2009.2013015. [15] H. Q. Dinh, Constacyclic codes of length $p^s$ over $\mathbb F_{p^m}+u\mathbb F_{p^m}$, J. Algebra, 324 (2010), 940-950. 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), 1728-1744. doi: 10.1109/TIT.2004.831789. [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), 32-45. doi: 10.1109/18.746770. [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), 326-341. [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), 2522-2524. doi: 10.1109/18.796395. [20] M. Greferath and S. E. Schmidt, Finite ring combinatorics and MacWilliams's equivalence theorem, J. Combin. Theory Ser. A, 92 (2000), 17-28. doi: 10.1006/jcta.1999.3033. [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), 301-319. doi: 10.1109/18.312154. [22] W. Heise, T. Honold and A. A. Nechaev, Weighted modules and representations of codes, in "Proceedings of the ACCT 6,'' Pskov, Russia, (1998), 123-129. [23] T. Honold and I. Landjev, Linear representable codes over chain rings, in "Proceedings of the ACCT 6,'' Pskov, Russia, (1998), 135-141. [24] W. C. Huffman and V. Pless, "Fundamentals of Error-Correcting Codes,'' Cambridge University Press, Cambridge, 2003. [25] S. Ling and P. Solé, Duadic codes over $\mathbb F_2+u\mathbb F_2$, Appl. Algebra Engrg. Comm. Comput., 12 (2001), 365-379. doi: 10.1007/s002000100079. [26] F. J. MacWilliams, Error-correcting codes for multiple-level transmissions, Bell System Tech. J., 40 (1961), 281-308. [27] F. J. MacWilliams, Combinatorial problems of elementary abelian groups, Ph.D thesis, Harvard University, Cambridge, MA, 1962. [28] J. L. Massey, D. J. Costello and J. Justesen, Polynomial weights and code constructions, IEEE Trans. Inform. Theory, 19 (1973), 101-110. doi: 10.1109/TIT.1973.1054936. [29] B. R. McDonald, "Finite Rings with Identity,'' Marcel Dekker, New York, 1974. [30] A. A. Nechaev, Kerdock code in a cyclic form (in Russian), Diskr. Math. (USSR), 1 (1989), 123-139; English translation: Discrete Math. Appl., 1 (1991), 365-384. doi: 10.1515/dma.1991.1.4.365. [31] C.-S. Nedeloaia, Weight distributions of cyclic self-dual codes, IEEE Trans. Inform. Theory, 49 (2003), 1582-1591. doi: 10.1109/TIT.2003.811921. [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), 489-506. doi: 10.1007/PL00012382. [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), 17-29. doi: 10.1007/s10623-004-5658-5. [34] V. Pless and W. C. Huffman, "Handbook of Coding Theory,'' Elsevier, Amsterdam, 1998. [35] R. M. Roth and G. Seroussi, On cyclic MDS codes of length $q$ over $\GF(q)$, IEEE Trans. Inform. Theory, 32 (1986), 284-285. doi: 10.1109/TIT.1986.1057151. [36] A. Sălăgean, Repeated-root cyclic and negacyclic codes over finite chain rings, Discrete Appl. Math., 154 (2006), 413-419. doi: 10.1016/j.dam.2005.03.016. [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), 732-737. doi: 10.1109/18.556131. [38] J. H. van Lint, Repeated-root cyclic codes, IEEE Trans. Inform. Theory, 37 (1991), 343-345. doi: 10.1109/18.75250. [39] J. A. Wood, Duality for modules over finite rings and applications to coding theory, American J. Math., 121 (1999), 555-575. doi: 10.1353/ajm.1999.0024. [40] K.-H. Zimmermann, On generalizations of repeated-root cyclic codes, IEEE Trans. Inform. Theory, 42 (1996), 641-649. doi: 10.1109/18.485736.
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