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On codes over 𝓡k, m and constructions for new binary self-dual codes

  • Nesibe Tufekci and Bahattin Yildiz EMAIL logo
Published/Copyright: December 30, 2016
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Mathematica Slovaca
From the journal Mathematica Slovaca Volume 66 Issue 6

Abstract

In this work, we study codes over the ring 𝓡k, m = 𝔽2[u, v]/ 〈uk, vm, uv − vu〉, which is a family of Frobenius, characteristic 2 extensions of the binary field. We introduce a distance and duality preserving Gray map from 𝓡k, m to F2km together with a Lee weight. After proving the MacWilliams identities for codes over 𝓡k, m for all the relevant weight enumerators, we construct many binary self-dual codes as the Gray images of self-dual codes over 𝓡k, m. In addition to many extremal binary self-dual codes obtained in this way, including a new construction for the extended binary Golay code, we find 175 new Type I binary self-dual codes of parameters [72, 36, 12] and 105 new Type II binary self-dual codes of parameter [72, 36, 12].

MSC 2010: Primary 94B05; Secondary 94B99
Keywords: extremal self-dual codes; Gray maps; codes over rings; MacWilliams identities

(Communicated by Federico Pellarin)

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Received: 2014-6-20
Accepted: 2014-11-25
Published Online: 2016-12-30
Published in Print: 2016-12-1

© 2016 Mathematical Institute Slovak Academy of Sciences

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https://doi.org/10.1515/ms-2016-0240
Keywords for this article
extremal self-dual codes; Gray maps; codes over rings; MacWilliams identities

Articles in the same Issue

  2. A triple representation of lattice effect algebras
  4. Periods of morgan-voyce sequences and elliptic curves
  6. Multiplicative generalized derivations on ideals in semiprime rings
  8. A few remarks on Poincaré-Perron solutions and regularly varying solutions
  10. Applications of extremal theorem and radius equation for a class of analytic functions
  12. On the “bang-bang” principle for a class of Riemann-Liouville fractional semilinear evolution inclusions
  14. New criteria for global exponential stability of linear time-varying volterra difference equations
  16. On approximation of functions by some hump matrix means of Fourier series
  18. Pseudo-amenability and pseudo-contractibility for certain products of Banach algebras
  20. Poisson kernels on semi-direct products of abelian groups
  22. Locally convex projective limit cones
  24. On the properties (wL) and (wV)
  26. On the existence of solutions for quadratic integral equations in Orlicz spaces
  28. Existence and uniqueness of best proximity points under rational contractivity conditions
  30. Almost Weyl structures on null geometry in indefinite Kenmotsu manifolds
  32. On the internal approach to differential equations 3. Infinitesimal symmetries
  34. A Characterization of the discontinuity point set of strongly separately continuous functions on products
  36. Wick differential and Poisson equations associated to the 𝚀𝚆𝙽-Euler operator acting on generalized operators
  38. Multivariate EIV models
  40. On codes over 𝓡k, m and constructions for new binary self-dual codes
  42. Domination number of total graphs
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