• 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).  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

[1]

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

[2]

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

[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]

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

[5]

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

[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 - A, 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]

Yanqin Xiong, Maoan Han. Planar quasi-homogeneous polynomial systems with a given weight degree. Discrete & Continuous Dynamical Systems - A, 2016, 36 (7) : 4015-4025. doi: 10.3934/dcds.2016.36.4015

[11]

Shudi Yang, Xiangli Kong, Xueying Shi. Complete weight enumerators of a class of linear codes over finite fields. Advances in Mathematics of Communications, 2019, 0 (0) : 0-0. doi: 10.3934/amc.2020045

[12]

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

[13]

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

[14]

Anderson Silva, C. Polcino Milies. Cyclic codes of length $ 2p^n $ over finite chain rings. Advances in Mathematics of Communications, 2019, 0 (0) : 0-0. doi: 10.3934/amc.2020017

[15]

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

[16]

Michal Kupsa, Štěpán Starosta. On the partitions with Sturmian-like refinements. Discrete & Continuous Dynamical Systems - A, 2015, 35 (8) : 3483-3501. doi: 10.3934/dcds.2015.35.3483

[17]

Bernard Helffer, Thomas Hoffmann-Ostenhof, Susanna Terracini. Nodal minimal partitions in dimension $3$. Discrete & Continuous Dynamical Systems - A, 2010, 28 (2) : 617-635. doi: 10.3934/dcds.2010.28.617

[18]

Xing-Fu Zhong. Variational principles of invariance pressures on partitions. Discrete & Continuous Dynamical Systems - A, 2020, 40 (1) : 491-508. doi: 10.3934/dcds.2020019

[19]

Stefan Klus, Péter Koltai, Christof Schütte. On the numerical approximation of the Perron-Frobenius and Koopman operator. Journal of Computational Dynamics, 2016, 3 (1) : 51-79. doi: 10.3934/jcd.2016003

[20]

Ren-You Zhong, Nan-Jing Huang. Strict feasibility for generalized mixed variational inequality in reflexive Banach spaces. Numerical Algebra, Control & Optimization, 2011, 1 (2) : 261-274. doi: 10.3934/naco.2011.1.261

2018 Impact Factor: 0.879

Metrics

  • PDF downloads (8)
  • HTML views (0)
  • Cited by (4)

Other articles
by authors

[Back to Top]