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Problems of Information Transmission

, Volume 42, Issue 1, pp 10–29 | Cite as

Vasil’ev codes of length n = 2m and doubling of Steiner systems S(n, 4, 3) of a given rank

  • V. A. Zinoviev
  • D. V. Zinoviev
Coding Theory
  • 33 Downloads

Abstract

Extended binary perfect nonlinear Vasil’ev codes of length n = 2m and Steiner systems S(n, 4, 3) of rank n-m over F 2 are studied. The generalized concatenated construction of Vasil’ev codes induces a variant of the doubling construction for Steiner systems S(n, 4, 3) of an arbitrary rank r over F 2. We prove that any Steiner system S(n = 2m, 4, 3) of rank n-m can be obtained by this doubling construction and is formed by codewords of weight 4 of these Vasil’ev codes. The length 16 is studied in detail. Orders of the full automorphism groups of all 12 nonequivalent Vasil’ev codes of length 16 are found. There are exactly 15 nonisomorphic systems S(16, 4, 3) of rank 12 over F 2, and they can be obtained from codewords of weight 4 of the extended Vasil’ev codes. Orders of the automorphism groups of all these Steiner systems are found.

Keywords

Vasil Information Transmission Double Coset Parity Rule Steiner Triple System 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Pleiades Publishing, Inc. 2006

Authors and Affiliations

  • V. A. Zinoviev
    • 1
  • D. V. Zinoviev
    • 1
  1. 1.Institute for Information Transmission ProblemsMoscowRussia

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