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May  2014, 8(2): 191-207. doi: 10.3934/amc.2014.8.191

Partitions of Frobenius rings induced by the homogeneous weight

1. 

University of Kentucky, Department of Mathematics, Lexington, KY 40506-0027, United States

Received  April 2013 Published  May 2014

The values of the homogeneous weight are determined for finite Frobenius rings that are a direct product of local Frobenius rings. This is used to investigate the partition induced by this weight and its dual partition under character-theoretic dualization. A characterization is given of those rings for which the induced partition is reflexive or even self-dual.
Citation: Heide Gluesing-Luerssen. Partitions of Frobenius rings induced by the homogeneous weight. Advances in Mathematics of Communications, 2014, 8 (2) : 191-207. doi: 10.3934/amc.2014.8.191
References:
[1]

A. Barra and H. Gluesing-Luerssen, MacWilliams extension theorems and the local-global property for codes over Frobenius rings, J. Pure Appl. Algebra, (2014), published online. doi: 10.1016/j.jpaa.2014.04.026.  Google Scholar

[2]

E. Byrne, On the weight distribution of codes over finite rings, Adv. Math. Commun., 5 (2011), 395-406. doi: 10.3934/amc.2011.5.395.  Google Scholar

[3]

E. Byrne, M. Greferath and M. E. O'Sullivan, The linear programming bound for codes over finite Frobenius rings, Des. Codes Cryptogr., 42 (2007), 289-301. doi: 10.1007/s10623-006-9035-4.  Google Scholar

[4]

E. Byrne, M. Kiermaier and A. Sneyd, Properties of codes with two homogeneous weights, Finite Fields Appl., 18 (2012), 711-727. doi: 10.1016/j.ffa.2012.01.002.  Google Scholar

[5]

P. Camion, Codes and association schemes, in Handbook of Coding Theory (eds. V.S. Pless and W.C. Huffman), Elsevier, 1998, 1441-1566.  Google Scholar

[6]

H. L. Claasen and R. W. Goldbach, A field-like property of finite rings, Indag. Math., 3 (1992), 11-26. doi: 10.1016/0019-3577(92)90024-F.  Google Scholar

[7]

I. Constantinescu and W. Heise, A metric for codes over residue class rings, Problems Inform. Transm., 33 (1997), 208-213.  Google Scholar

[8]

I. Constantinescu, W. Heise and T. Honold, Monomial extensions of isometries between codes over $\mathbb Z_m$, in Proceedings of the Fifth International Workshop on Algebraic and Combinatorial Coding Theory (ACCT '96), Unicorn, Shumen, 1996, 98-104. Google Scholar

[9]

P. Delsarte, An Algebraic Approach to the Association Schemes of Coding Theory, Ph.D thesis, Universite Catholique de Louvain, 1973.  Google Scholar

[10]

P. Delsarte and V. Levenshtein, Association schemes and coding theory, IEEE Trans. Inform. Theory, IT-44 (1998), 2477-2504. doi: 10.1109/18.720545.  Google Scholar

[11]

I. M. Duursma, M. Greferath, S. N. Litsyn and S. E. Schmidt, A $\mathbb Z_8$-linear shift of the binary Golay code and a nonlinear binary $(96,2^{37},24)$-code, IEEE Trans. Inform. Theory, IT-47 (2001), 1596-1598. doi: 10.1109/18.923742.  Google Scholar

[12]

Y. Fan, S. Ling and H. Liu, Homogeneous weights of matrix product codes over finite principal ideal rings,, preprint, ().   Google Scholar

[13]

Y. Fan and H. Liu, Homogeneous weights of finite rings and Möbius functions, Math. Ann. (Chinese), 31A (2010), 355-364. English translation: arXiv:1304.4927  Google Scholar

[14]

H. Gluesing-Luerssen, Fourier-reflexive partitions and MacWilliams identities for additive codes, Des. Codes Cryptogr., (2014), published online. doi: 10.1007/s10623-014-9940-x.  Google Scholar

[15]

C. D. Godsil, Algebraic Combinatorics, Chapman and Hall, New York, 1993.  Google Scholar

[16]

M. Greferath, A. Nechaev and R. Wisbauer, Finite quasi-Frobenius modules and linear codes, J. Algebra Appl., 3 (2000), 247-272. doi: 10.1142/S0219498804000873.  Google Scholar

[17]

M. Greferath and M. E. O'Sullivan, On bounds for codes over Frobenius rings under homogeneous weights, Discrete Math., 289 (2004), 11-24. doi: 10.1016/j.disc.2004.10.002.  Google Scholar

[18]

M. Greferath and S. E. Schmidt, Finite ring combinatorics and MacWilliams' equivalence theorem, J. Combin. Theory Ser. A, 92 (2000), 17-28. doi: 10.1006/jcta.1999.3033.  Google Scholar

[19]

Y. Hirano, On admissible rings, Indag. Math., 8 (1997), 55-59. doi: 10.1016/S0019-3577(97)83350-2.  Google Scholar

[20]

T. Honold, Characterization of finite Frobenius rings, Arch. Math., 76 (2001), 406-415. doi: 10.1007/PL00000451.  Google Scholar

[21]

T. Honold, Two-intersection sets in projective Hjelmslev spaces, in Proceedings of the 19th International Symposium on the Mathematical Theory of Networks and Systems, Budapest, 2010, 1807-1813. Google Scholar

[22]

T. Honold and I. Landjev, MacWilliams identities for linear codes over finite Frobenius rings, in Finite Fields and Applications (eds. D. Jungnickel and H. Niederreiter), Springer, 2001, 276-292.  Google Scholar

[23]

T. Honold and A. A. Nechaev, Weighted modules and linear representations of codes, Problems Inform. Transm., 35 (1999), 205-223.  Google Scholar

[24]

W. C. Huffman and V. Pless, Fundamental of Error-Correcting Codes, Cambridge Univ. Press, Cambridge, 2003. doi: 10.1017/CBO9780511807077.  Google Scholar

[25]

T. Y. Lam, Lectures on Modules and Rings, Springer, 1999. doi: 10.1007/978-1-4612-0525-8.  Google Scholar

[26]

E. Lamprecht, Über I-reguläre Ringe, reguläre Ideale and Erklärungsmoduln, I. Math. Nachr., 10 (1953), 353-382.  Google Scholar

[27]

J. F. Voloch and J. L. Walker, Homogeneous weights and exponential sums, Finite Fields Appl., 9 (2003), 310-321. doi: 10.1016/S1071-5797(03)00007-8.  Google Scholar

[28]

J. A. Wood, Extension theorems for linear codes over finite rings, in Applied Algebra, Algebraic Algorithms and Error-Correcting Codes (eds. T. Mora and H. Mattson), Springer, 1997, 329-340. doi: 10.1007/3-540-63163-1_26.  Google Scholar

[29]

J. A. Wood, Duality for modules over finite rings and applications to coding theory, Americ. J. Math., 121 (1999), 555-575.  Google Scholar

[30]

V. A. Zinoviev and T. Ericson, On Fourier invariant partitions of finite abelian groups and the MacWilliams identity for group codes, Problems Inform. Transm., 32 (1996), 117-122.  Google Scholar

[31]

V. A. Zinoviev and T. Ericson, Fourier invariant pairs of partitions of finite abelian groups and association schemes, Problems Inform. Transm., 45 (2009), 221-231. doi: 10.1134/S003294600903003X.  Google Scholar

show all references

References:
[1]

A. Barra and H. Gluesing-Luerssen, MacWilliams extension theorems and the local-global property for codes over Frobenius rings, J. Pure Appl. Algebra, (2014), published online. doi: 10.1016/j.jpaa.2014.04.026.  Google Scholar

[2]

E. Byrne, On the weight distribution of codes over finite rings, Adv. Math. Commun., 5 (2011), 395-406. doi: 10.3934/amc.2011.5.395.  Google Scholar

[3]

E. Byrne, M. Greferath and M. E. O'Sullivan, The linear programming bound for codes over finite Frobenius rings, Des. Codes Cryptogr., 42 (2007), 289-301. doi: 10.1007/s10623-006-9035-4.  Google Scholar

[4]

E. Byrne, M. Kiermaier and A. Sneyd, Properties of codes with two homogeneous weights, Finite Fields Appl., 18 (2012), 711-727. doi: 10.1016/j.ffa.2012.01.002.  Google Scholar

[5]

P. Camion, Codes and association schemes, in Handbook of Coding Theory (eds. V.S. Pless and W.C. Huffman), Elsevier, 1998, 1441-1566.  Google Scholar

[6]

H. L. Claasen and R. W. Goldbach, A field-like property of finite rings, Indag. Math., 3 (1992), 11-26. doi: 10.1016/0019-3577(92)90024-F.  Google Scholar

[7]

I. Constantinescu and W. Heise, A metric for codes over residue class rings, Problems Inform. Transm., 33 (1997), 208-213.  Google Scholar

[8]

I. Constantinescu, W. Heise and T. Honold, Monomial extensions of isometries between codes over $\mathbb Z_m$, in Proceedings of the Fifth International Workshop on Algebraic and Combinatorial Coding Theory (ACCT '96), Unicorn, Shumen, 1996, 98-104. Google Scholar

[9]

P. Delsarte, An Algebraic Approach to the Association Schemes of Coding Theory, Ph.D thesis, Universite Catholique de Louvain, 1973.  Google Scholar

[10]

P. Delsarte and V. Levenshtein, Association schemes and coding theory, IEEE Trans. Inform. Theory, IT-44 (1998), 2477-2504. doi: 10.1109/18.720545.  Google Scholar

[11]

I. M. Duursma, M. Greferath, S. N. Litsyn and S. E. Schmidt, A $\mathbb Z_8$-linear shift of the binary Golay code and a nonlinear binary $(96,2^{37},24)$-code, IEEE Trans. Inform. Theory, IT-47 (2001), 1596-1598. doi: 10.1109/18.923742.  Google Scholar

[12]

Y. Fan, S. Ling and H. Liu, Homogeneous weights of matrix product codes over finite principal ideal rings,, preprint, ().   Google Scholar

[13]

Y. Fan and H. Liu, Homogeneous weights of finite rings and Möbius functions, Math. Ann. (Chinese), 31A (2010), 355-364. English translation: arXiv:1304.4927  Google Scholar

[14]

H. Gluesing-Luerssen, Fourier-reflexive partitions and MacWilliams identities for additive codes, Des. Codes Cryptogr., (2014), published online. doi: 10.1007/s10623-014-9940-x.  Google Scholar

[15]

C. D. Godsil, Algebraic Combinatorics, Chapman and Hall, New York, 1993.  Google Scholar

[16]

M. Greferath, A. Nechaev and R. Wisbauer, Finite quasi-Frobenius modules and linear codes, J. Algebra Appl., 3 (2000), 247-272. doi: 10.1142/S0219498804000873.  Google Scholar

[17]

M. Greferath and M. E. O'Sullivan, On bounds for codes over Frobenius rings under homogeneous weights, Discrete Math., 289 (2004), 11-24. doi: 10.1016/j.disc.2004.10.002.  Google Scholar

[18]

M. Greferath and S. E. Schmidt, Finite ring combinatorics and MacWilliams' equivalence theorem, J. Combin. Theory Ser. A, 92 (2000), 17-28. doi: 10.1006/jcta.1999.3033.  Google Scholar

[19]

Y. Hirano, On admissible rings, Indag. Math., 8 (1997), 55-59. doi: 10.1016/S0019-3577(97)83350-2.  Google Scholar

[20]

T. Honold, Characterization of finite Frobenius rings, Arch. Math., 76 (2001), 406-415. doi: 10.1007/PL00000451.  Google Scholar

[21]

T. Honold, Two-intersection sets in projective Hjelmslev spaces, in Proceedings of the 19th International Symposium on the Mathematical Theory of Networks and Systems, Budapest, 2010, 1807-1813. Google Scholar

[22]

T. Honold and I. Landjev, MacWilliams identities for linear codes over finite Frobenius rings, in Finite Fields and Applications (eds. D. Jungnickel and H. Niederreiter), Springer, 2001, 276-292.  Google Scholar

[23]

T. Honold and A. A. Nechaev, Weighted modules and linear representations of codes, Problems Inform. Transm., 35 (1999), 205-223.  Google Scholar

[24]

W. C. Huffman and V. Pless, Fundamental of Error-Correcting Codes, Cambridge Univ. Press, Cambridge, 2003. doi: 10.1017/CBO9780511807077.  Google Scholar

[25]

T. Y. Lam, Lectures on Modules and Rings, Springer, 1999. doi: 10.1007/978-1-4612-0525-8.  Google Scholar

[26]

E. Lamprecht, Über I-reguläre Ringe, reguläre Ideale and Erklärungsmoduln, I. Math. Nachr., 10 (1953), 353-382.  Google Scholar

[27]

J. F. Voloch and J. L. Walker, Homogeneous weights and exponential sums, Finite Fields Appl., 9 (2003), 310-321. doi: 10.1016/S1071-5797(03)00007-8.  Google Scholar

[28]

J. A. Wood, Extension theorems for linear codes over finite rings, in Applied Algebra, Algebraic Algorithms and Error-Correcting Codes (eds. T. Mora and H. Mattson), Springer, 1997, 329-340. doi: 10.1007/3-540-63163-1_26.  Google Scholar

[29]

J. A. Wood, Duality for modules over finite rings and applications to coding theory, Americ. J. Math., 121 (1999), 555-575.  Google Scholar

[30]

V. A. Zinoviev and T. Ericson, On Fourier invariant partitions of finite abelian groups and the MacWilliams identity for group codes, Problems Inform. Transm., 32 (1996), 117-122.  Google Scholar

[31]

V. A. Zinoviev and T. Ericson, Fourier invariant pairs of partitions of finite abelian groups and association schemes, Problems Inform. Transm., 45 (2009), 221-231. doi: 10.1134/S003294600903003X.  Google Scholar

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