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

This article is cited in 33 scientific papers (total in 33 papers)

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 Mцbius 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.

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

Bibliographic databases:
UDC: 621.391.15
Received: 09.06.1998

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


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    Citing articles on Google Scholar: Russian citations, English citations
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    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  mathnet  crossref  crossref  mathscinet  zmath
    2. Honold, T, “Characterizations of finite Frobenius rings”, Archiv der Mathematik, 76:6 (2001), 406  crossref  mathscinet  zmath  isi
    3. Honold T., Landjev I., “MacWilliams identities for linear codes over finite Frobenius rings”, Finite Fields and Applications, 2001, 276–292  crossref  mathscinet  zmath  isi
    4. Greferath, M, “Orthogonality matrices for modules over finite Frobenius rings and MacWilliams' equivalence theorem”, Finite Fields and Their Applications, 8:3 (2002), 323  crossref  mathscinet  zmath  isi
    5. Voloch, JF, “Homogeneous weights and exponential sums”, Finite Fields and Their Applications, 9:3 (2003), 310  crossref  mathscinet  zmath  isi
    6. Muniz M., Costa S.I.R., “Labelings of Lee and Hamming Spaces”, Discrete Math, 260:1–3 (2003), 119–136  crossref  mathscinet  zmath  isi
    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  mathnet  crossref  mathscinet  zmath
    8. Krotov, DS, “On Z(2)k-dual binary codes”, IEEE Transactions on Information Theory, 53:4 (2007), 1532  crossref  mathscinet  zmath  isi
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    10. Ozbudak, F, “Gilbert-Varshamov type bounds for linear codes over finite chain rings”, Advances in Mathematics of Communications, 1:1 (2007), 99  crossref  mathscinet  zmath  isi
    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  crossref  mathscinet  isi
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    14. Byrne, E, “Ring geometries, two-weight codes, and strongly regular graphs”, Designs Codes and Cryptography, 48:1 (2008), 1  crossref  mathscinet  zmath  isi
    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  mathscinet  zmath  isi
    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  mathscinet  zmath  isi
    17. Greferath M. Nechaev A., “Generalized Frobenius Extensions of Finite Rings and Trace Functions”, 2010 IEEE Information Theory Workshop (Itw), IEEE, 2010  isi
    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  crossref  mathscinet  zmath  isi
    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  crossref  mathscinet  zmath  isi
    20. Byrne E. Kiermaier M. Sneyd A., “Properties of Codes with Two Homogeneous Weights”, Finite Fields their Appl., 18:4 (2012), 711–727  crossref  mathscinet  zmath  isi  elib
    21. Yang Sh. Honold T., “Good Random Matrices Over Finite Fields”, Adv. Math. Commun., 6:2 (2012), 203–227  crossref  mathscinet  zmath  isi  elib
    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  crossref  mathscinet  zmath  isi  elib
    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  crossref  mathscinet  zmath  isi
    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  crossref  mathscinet  zmath  isi  elib
    25. Krotov D.S., Potapov V.N., “Propelinear 1-Perfect Codes From Quadratic Functions”, IEEE Trans. Inf. Theory, 60:4 (2014), 2065–2068  crossref  mathscinet  isi  elib
    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  crossref  mathscinet  zmath  isi
    27. Wood J.A., “Relative One-Weight Linear Codes”, Des. Codes Cryptogr., 72:2 (2014), 331–344  crossref  mathscinet  zmath  isi
    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  crossref  mathscinet  zmath  isi
    29. Gluesing-Luerssen H., “Partitions of Frobenius Rings Induced By the Homogeneous Weight”, Adv. Math. Commun., 8:2 (2014), 191–207  crossref  mathscinet  zmath  isi  elib
    30. Gluesing-Luerssen H., “the Homogeneous Weight Partition and Its Character-Theoretic Dual”, Des. Codes Cryptogr., 79:1 (2016), 47–61  crossref  mathscinet  zmath  isi  elib
    31. Westerback T., “Parity Check Systems of Nonlinear Codes Over Finite Commutative Frobenius Rings”, Adv. Math. Commun., 11:3 (2017), 409–427  crossref  mathscinet  zmath  isi  scopus
    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  crossref  mathscinet  isi
    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  crossref  mathscinet  zmath  isi  scopus
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