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

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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.

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  1. Institute for Information Transmission Problems, Moscow, Russia

    V. A. Zinoviev & D. V. Zinoviev

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  1. V. A. Zinoviev
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  2. D. V. Zinoviev
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Original Russian Text © V.A. Zinoviev, D.V. Zinoviev, 2006, published in Problemy Peredachi Informatsii, 2006, Vol. 42, No. 1, pp. 13–33.

Supported in part by the Russian Foundation for Basic Research, project no. 03-01-00098.

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Zinoviev, V.A., Zinoviev, D.V. Vasil’ev codes of length n = 2m and doubling of Steiner systems S(n, 4, 3) of a given rank. Probl Inf Transm 42, 10–29 (2006). https://doi.org/10.1134/S0032946006010029

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  • DOI: https://doi.org/10.1134/S0032946006010029

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