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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). 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. 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. 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. 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), (1998), 1441. Google Scholar [6] H. L. Claasen and R. W. Goldbach, A field-like property of finite rings,, Indag. Math., 3 (1992), 11. 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. 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), (1996), 98. Google Scholar [9] P. Delsarte, An Algebraic Approach to the Association Schemes of Coding Theory,, Ph.D thesis, (1973). Google Scholar [10] P. Delsarte and V. Levenshtein, Association schemes and coding theory,, IEEE Trans. Inform. Theory, IT-44 (1998), 2477. 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. 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. Google Scholar [14] H. Gluesing-Luerssen, Fourier-reflexive partitions and MacWilliams identities for additive codes,, Des. Codes Cryptogr., (2014). doi: 10.1007/s10623-014-9940-x. Google Scholar [15] C. D. Godsil, Algebraic Combinatorics,, Chapman and Hall, (1993). Google Scholar [16] M. Greferath, A. Nechaev and R. Wisbauer, Finite quasi-Frobenius modules and linear codes,, J. Algebra Appl., 3 (2000), 247. 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. 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. doi: 10.1006/jcta.1999.3033. Google Scholar [19] Y. Hirano, On admissible rings,, Indag. Math., 8 (1997), 55. doi: 10.1016/S0019-3577(97)83350-2. Google Scholar [20] T. Honold, Characterization of finite Frobenius rings,, Arch. Math., 76 (2001), 406. 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, (2010), 1807. 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), (2001), 276. Google Scholar [23] T. Honold and A. A. Nechaev, Weighted modules and linear representations of codes,, Problems Inform. Transm., 35 (1999), 205. Google Scholar [24] W. C. Huffman and V. Pless, Fundamental of Error-Correcting Codes,, Cambridge Univ. Press, (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. Google Scholar [27] J. F. Voloch and J. L. Walker, Homogeneous weights and exponential sums,, Finite Fields Appl., 9 (2003), 310. 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, (1997), 329. 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. 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. 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. 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). 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. 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. 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. 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), (1998), 1441. Google Scholar [6] H. L. Claasen and R. W. Goldbach, A field-like property of finite rings,, Indag. Math., 3 (1992), 11. 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. 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), (1996), 98. Google Scholar [9] P. Delsarte, An Algebraic Approach to the Association Schemes of Coding Theory,, Ph.D thesis, (1973). Google Scholar [10] P. Delsarte and V. Levenshtein, Association schemes and coding theory,, IEEE Trans. Inform. Theory, IT-44 (1998), 2477. 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. 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. Google Scholar [14] H. Gluesing-Luerssen, Fourier-reflexive partitions and MacWilliams identities for additive codes,, Des. Codes Cryptogr., (2014). doi: 10.1007/s10623-014-9940-x. Google Scholar [15] C. D. Godsil, Algebraic Combinatorics,, Chapman and Hall, (1993). Google Scholar [16] M. Greferath, A. Nechaev and R. Wisbauer, Finite quasi-Frobenius modules and linear codes,, J. Algebra Appl., 3 (2000), 247. 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. 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. doi: 10.1006/jcta.1999.3033. Google Scholar [19] Y. Hirano, On admissible rings,, Indag. Math., 8 (1997), 55. doi: 10.1016/S0019-3577(97)83350-2. Google Scholar [20] T. Honold, Characterization of finite Frobenius rings,, Arch. Math., 76 (2001), 406. 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, (2010), 1807. 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), (2001), 276. Google Scholar [23] T. Honold and A. A. Nechaev, Weighted modules and linear representations of codes,, Problems Inform. Transm., 35 (1999), 205. Google Scholar [24] W. C. Huffman and V. Pless, Fundamental of Error-Correcting Codes,, Cambridge Univ. Press, (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. Google Scholar [27] J. F. Voloch and J. L. Walker, Homogeneous weights and exponential sums,, Finite Fields Appl., 9 (2003), 310. 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, (1997), 329. 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. 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. 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. doi: 10.1134/S003294600903003X. Google Scholar
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