On the Steiner quadruple systems of small rank which are embeddable into extended perfect binary codes

Abstract

The codewords of weight 4 of every extended perfect binary code that contains the all- zero vector are known to form a Steiner quadruple system. We propose a modification of the Lindner construction for the Steiner quadruple system of order N =2 r which can be described by special switchings from the Hamming Steiner quadruple system. We prove that each of these Steiner quadruple systems is embedded into some extended perfect binary code constructed by the method of switching of ijkl-components from the binary extended Hamming code. We give the lower bound for the number of different Steiner quadruple systems of order N with rank at most N − log N +1 which are embedded into extended perfect codes of length N.