Vasil'ev Codes of Length $n=2^m$ and Doubling of Steiner Systems $S(n,4,3)$ of a Given Rank

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.