Steiner quadruple systems of small rank embedded into extended perfect binary codes

It is known that the set of all vectors of weight 4 in an arbitrary extended perfect binary code of length $N$ containing the all-zero vector defines a Steiner quadruple system of order $N$. In this paper, we give a modification of the known Lindner construction for the Steiner quadruple system of order $N=2^r$ that can be represented by some special switchings from the Hamming system of Steiner quadruples. It is proved that any of such Steiner quadruple systems is embedded into some extended perfect binary code constructed by switchings of $ijkl$-components from the binary extended Hamming code. We present the lower bound for the number of different Steiner quadruple systems of order $N$ of rank less than or equal to $N-\log N+1$ such that the systems are embedded into extended perfect binary codes of length $N$. Tab. 4, bibliogr. 19.