Abstract:
Extended binary perfect nonlinear Vasil'ev codes of length $n=2^m$ and Steiner
systems $S(n,4,3)$ of rank $n-m$ over $\mathbb F_2$ 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 $\mathbb F_2$. We prove that any Steiner system $S(n=2^m,4,3)$
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 $\mathbb F_2$, 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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