# American Institute of Mathematical Sciences

• Previous Article
A Fourier transform approach for improving the Levenshtein's lower bound on aperiodic correlation of binary sequences
• AMC Home
• This Issue
• Next Article
Algebraic space-time codes based on division algebras with a unitary involution
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
 [1] Thomas Westerbäck. Parity check systems of nonlinear codes over finite commutative Frobenius rings. Advances in Mathematics of Communications, 2017, 11 (3) : 409-427. doi: 10.3934/amc.2017035 [2] Eimear Byrne. On the weight distribution of codes over finite rings. Advances in Mathematics of Communications, 2011, 5 (2) : 395-406. doi: 10.3934/amc.2011.5.395 [3] Steven T. Dougherty, Abidin Kaya, Esengül Saltürk. Cyclic codes over local Frobenius rings of order 16. Advances in Mathematics of Communications, 2017, 11 (1) : 99-114. doi: 10.3934/amc.2017005 [4] Nuh Aydin, Yasemin Cengellenmis, Abdullah Dertli, Steven T. Dougherty, Esengül Saltürk. Skew constacyclic codes over the local Frobenius non-chain rings of order 16. Advances in Mathematics of Communications, 2020, 14 (1) : 53-67. doi: 10.3934/amc.2020005 [5] Sergio R. López-Permouth, Steve Szabo. On the Hamming weight of repeated root cyclic and negacyclic codes over Galois rings. Advances in Mathematics of Communications, 2009, 3 (4) : 409-420. doi: 10.3934/amc.2009.3.409 [6] Aicha Batoul, Kenza Guenda, T. Aaron Gulliver. Some constacyclic codes over finite chain rings. Advances in Mathematics of Communications, 2016, 10 (4) : 683-694. doi: 10.3934/amc.2016034 [7] Somphong Jitman, San Ling, Patanee Udomkavanich. Skew constacyclic codes over finite chain rings. Advances in Mathematics of Communications, 2012, 6 (1) : 39-63. doi: 10.3934/amc.2012.6.39 [8] Igor E. Shparlinski. On some dynamical systems in finite fields and residue rings. Discrete & Continuous Dynamical Systems, 2007, 17 (4) : 901-917. doi: 10.3934/dcds.2007.17.901 [9] M. DeDeo, M. Martínez, A. Medrano, M. Minei, H. Stark, A. Terras. Spectra of Heisenberg graphs over finite rings. Conference Publications, 2003, 2003 (Special) : 213-222. doi: 10.3934/proc.2003.2003.213 [10] Yue Feng, Yujie Liu, Ruishu Wang, Shangyou Zhang. A conforming discontinuous Galerkin finite element method on rectangular partitions. Electronic Research Archive, 2021, 29 (3) : 2375-2389. doi: 10.3934/era.2020120 [11] Yanqin Xiong, Maoan Han. Planar quasi-homogeneous polynomial systems with a given weight degree. Discrete & Continuous Dynamical Systems, 2016, 36 (7) : 4015-4025. doi: 10.3934/dcds.2016.36.4015 [12] Shudi Yang, Xiangli Kong, Xueying Shi. Complete weight enumerators of a class of linear codes over finite fields. Advances in Mathematics of Communications, 2021, 15 (1) : 99-112. doi: 10.3934/amc.2020045 [13] David Grant, Mahesh K. Varanasi. The equivalence of space-time codes and codes defined over finite fields and Galois rings. Advances in Mathematics of Communications, 2008, 2 (2) : 131-145. doi: 10.3934/amc.2008.2.131 [14] Anderson Silva, C. Polcino Milies. Cyclic codes of length $2p^n$ over finite chain rings. Advances in Mathematics of Communications, 2020, 14 (2) : 233-245. doi: 10.3934/amc.2020017 [15] Ferruh Özbudak, Patrick Solé. Gilbert-Varshamov type bounds for linear codes over finite chain rings. Advances in Mathematics of Communications, 2007, 1 (1) : 99-109. doi: 10.3934/amc.2007.1.99 [16] Virginie Bonnaillie-Noël, Corentin Léna. Spectral minimal partitions of a sector. Discrete & Continuous Dynamical Systems - B, 2014, 19 (1) : 27-53. doi: 10.3934/dcdsb.2014.19.27 [17] Xing-Fu Zhong. Variational principles of invariance pressures on partitions. Discrete & Continuous Dynamical Systems, 2020, 40 (1) : 491-508. doi: 10.3934/dcds.2020019 [18] Michal Kupsa, Štěpán Starosta. On the partitions with Sturmian-like refinements. Discrete & Continuous Dynamical Systems, 2015, 35 (8) : 3483-3501. doi: 10.3934/dcds.2015.35.3483 [19] Bernard Helffer, Thomas Hoffmann-Ostenhof, Susanna Terracini. Nodal minimal partitions in dimension $3$. Discrete & Continuous Dynamical Systems, 2010, 28 (2) : 617-635. doi: 10.3934/dcds.2010.28.617 [20] Mónica Clapp, Juan Carlos Fernández, Alberto Saldaña. Critical polyharmonic systems and optimal partitions. Communications on Pure & Applied Analysis, 2021, 20 (11) : 4007-4023. doi: 10.3934/cpaa.2021141

2020 Impact Factor: 0.935

## Metrics

• HTML views (0)
• Cited by (5)

• on AIMS