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Combinatorial batch codes: A lower bound and optimal constructions
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 |
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. |
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), 3306-3312.
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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