RUS  ENG JOURNALS   PEOPLE   ORGANISATIONS   CONFERENCES   SEMINARS   VIDEO LIBRARY   PACKAGE AMSBIB
 General information Latest issue Archive Impact factor Guidelines for authors Search papers Search references RSS Latest issue Current issues Archive issues What is RSS

 Probl. Peredachi Inf.: Year: Volume: Issue: Page: Find

 Probl. Peredachi Inf., 1999, Volume 35, Issue 3, Pages 18–39 (Mi ppi450)

Coding Theory

Weighted Modules and Representations of Codes

A. A. Nechaev, T. Khonol'd

Abstract: A weight on a finite module is called egalitarian if the average weights of elements of any two its nonzero submodules are equal and it is called homogeneous if in addition the weights of any two associated elements are equal. The criteria of the existence of the egalitarian and homogeneous weights on an arbitrary finite module and the description of possible homogeneous weights are given. These results generalize the analogous results of Constantinescu and Heise for the ring $\mathbb Z_m$. Those finite modules which admit a homogeneous weight are called weighted and characterized in terms of the composition factors of their socle. A homogeneous weight in terms of Möbius and Euler functions for finite modules is described and effectively calculated. As an application, besides the known presentation of the generalized Kerdock code, also isometric representations of the Golay codes and the generalized Reed–Muller codes as short linear codes over modules are given.

Full text: PDF file (2605 kB)
References: PDF file   HTML file

English version:
Problems of Information Transmission, 1999, 35:3, 205–223

Bibliographic databases:
UDC: 621.391.15

Citation: A. A. Nechaev, T. Khonol'd, “Weighted Modules and Representations of Codes”, Probl. Peredachi Inf., 35:3 (1999), 18–39; Problems Inform. Transmission, 35:3 (1999), 205–223

Citation in format AMSBIB
\Bibitem{NecKho99} \by A.~A.~Nechaev, T.~Khonol'd \paper Weighted Modules and Representations of Codes \jour Probl. Peredachi Inf. \yr 1999 \vol 35 \issue 3 \pages 18--39 \mathnet{http://mi.mathnet.ru/ppi450} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=1730800} \zmath{https://zbmath.org/?q=an:1004.94028} \transl \jour Problems Inform. Transmission \yr 1999 \vol 35 \issue 3 \pages 205--223 

• http://mi.mathnet.ru/eng/ppi450
• http://mi.mathnet.ru/eng/ppi/v35/i3/p18

 SHARE:

Citing articles on Google Scholar: Russian citations, English citations
Related articles on Google Scholar: Russian articles, English articles

This publication is cited in the following articles:
1. I. N. Landjev, T. Khonol'd, “Arcs in projective Hjelmslev planes”, Discrete Math. Appl., 11:1 (2001), 53–70
2. Honold, T, “Characterizations of finite Frobenius rings”, Archiv der Mathematik, 76:6 (2001), 406
3. Honold T., Landjev I., “MacWilliams identities for linear codes over finite Frobenius rings”, Finite Fields and Applications, 2001, 276–292
4. Greferath, M, “Orthogonality matrices for modules over finite Frobenius rings and MacWilliams' equivalence theorem”, Finite Fields and Their Applications, 8:3 (2002), 323
5. Voloch, JF, “Homogeneous weights and exponential sums”, Finite Fields and Their Applications, 9:3 (2003), 310
6. Muniz M., Costa S.I.R., “Labelings of Lee and Hamming Spaces”, Discrete Math, 260:1–3 (2003), 119–136
7. D. V. Zinov'ev, P. Solé, “Quaternary Codes and Biphase Codes from $\mathbb Z_8$-Codes”, Problems Inform. Transmission, 40:2 (2004), 147–158
8. Krotov, DS, “On Z(2)k-dual binary codes”, IEEE Transactions on Information Theory, 53:4 (2007), 1532
9. Landjev, I, “On blocking sets in projective Hjelmslev planes”, Advances in Mathematics of Communications, 1:1 (2007), 65
10. Ozbudak, F, “Gilbert-Varshamov type bounds for linear codes over finite chain rings”, Advances in Mathematics of Communications, 1:1 (2007), 99
11. Sole, P, “Bounds on the minimum homogeneous distance of the p(r)-ary image of linear block codes over the Galois ring GR(p(r), m)”, IEEE Transactions on Information Theory, 53:6 (2007), 2270
12. Byrne, E, “The linear programming bound for codes over finite Frobenius rings”, Designs Codes and Cryptography, 42:3 (2007), 289
13. Sole P., Sison V., “Bounds on the minimum homogeneous distance of the p(r)-ary image of linear block codes over the Galois ring GR(p(r), m)”, 2007 IEEE International Symposium on Information Theory Proceedings, 2007, 1971–1974
14. Byrne, E, “Ring geometries, two-weight codes, and strongly regular graphs”, Designs Codes and Cryptography, 48:1 (2008), 1
15. Landjev, I, “A FAMILY OF TWO-WEIGHT RING CODES AND STRONGLY REGULAR GRAPHS”, Comptes Rendus de l Academie Bulgare Des Sciences, 62:3 (2009), 297
16. Honold T., Landjev I., “Linear Codes over Finite Chain Rings and Projective Hjelmslev Geometries”, Codes Over Rings, Series on Coding Theory and Cryptology, 6, 2009, 60–123
17. Greferath M. Nechaev A., “Generalized Frobenius Extensions of Finite Rings and Trace Functions”, 2010 IEEE Information Theory Workshop (Itw), IEEE, 2010
18. Kiermaier M., Zwanzger J., “A Z(4)-Linear Code of High Minimum Lee Distance Derived From a Hyperoval”, Adv Math Commun, 5:2 (2011), 275–286
19. Lopez-Andrade C.A., Tapia-Recillas H., “On the Linearity and Quasi-cyclicity of the Gray Image of Codes over a Galois Ring”, Groups, Algebras and Applications, Contemporary Mathematics, 537, 2011, 255–268
20. Byrne E. Kiermaier M. Sneyd A., “Properties of Codes with Two Homogeneous Weights”, Finite Fields their Appl., 18:4 (2012), 711–727
21. Yang Sh. Honold T., “Good Random Matrices Over Finite Fields”, Adv. Math. Commun., 6:2 (2012), 203–227
22. Kiermaier M. Zwanzger J., “New Ring-Linear Codes From Dualization in Projective Hjelmslev Geometries”, Des. Codes Cryptogr., 66:1-3, SI (2013), 39–55
23. Greferath M., Mc Fadden C., Zumbraegel J., “Characteristics of Invariant Weights Related to Code Equivalence Over Rings”, Des. Codes Cryptogr., 66:1-3, SI (2013), 145–156
24. Honold T. Kiermaier M., “The Existence of Maximal (Q (2), 2)-Arcs in Projective Hjelmslev Planes Over Chain Rings of Length 2 and Odd Prime Characteristic”, Des. Codes Cryptogr., 68:1-3, SI (2013), 105–126
25. Krotov D.S., Potapov V.N., “Propelinear 1-Perfect Codes From Quadratic Functions”, IEEE Trans. Inf. Theory, 60:4 (2014), 2065–2068
26. Fan Yu. Ling S. Liu H., “Homogeneous Weights of Matrix Product Codes Over Finite Principal Ideal Rings”, Finite Fields their Appl., 29 (2014), 247–267
27. Wood J.A., “Relative One-Weight Linear Codes”, Des. Codes Cryptogr., 72:2 (2014), 331–344
28. Greferath M. Honold T. Mc Fadden C. Wood J.A. Zumbraegel J., “Macwilliams' Extension Theorem For Bi-Invariant Weights Over Finite Principal Ideal Rings”, J. Comb. Theory Ser. A, 125 (2014), 177–193
29. Gluesing-Luerssen H., “Partitions of Frobenius Rings Induced By the Homogeneous Weight”, Adv. Math. Commun., 8:2 (2014), 191–207
30. Gluesing-Luerssen H., “the Homogeneous Weight Partition and Its Character-Theoretic Dual”, Des. Codes Cryptogr., 79:1 (2016), 47–61
31. Westerback T., “Parity Check Systems of Nonlinear Codes Over Finite Commutative Frobenius Rings”, Adv. Math. Commun., 11:3 (2017), 409–427
32. Shi M. Alahmadi A. Sole P., “Few Weight Codes”: Shi, M Alahmadi, A Sole, P, Codes and Rings: Theory and Practice, Pure and Applied Mathematics, Elsevier Academic Press Inc, 2017, 29–69
33. Byrne E., “Induced Weights on Quotient Modules and An Application to Error Correction in Coherent Networks”, Finite Fields their Appl., 52 (2018), 174–199
•  Number of views: This page: 776 Full text: 171 References: 41 First page: 2