ISSN 1990-4789, Journal of Applied and Industrial Mathematics, 2013, Vol. 7, No. 4, pp. 522–536. c  Pleiades Publishing, Ltd., 2013. Original Russian Text c  D.I. Kovalevskaya, F.I. Solov’eva, 2013, published in Diskretnyi Analiz i Issledovanie Operatsii, 2013, Vol. 20, No. 4, pp. 46–64. Steiner Quadruple Systems of Small Rank and Extended Perfect Binary Codes D. I. Kovalevskaya 1* and F. I. Solov’eva 1,2** 1 Sobolev Institute of Mathematics, pr. Akad. Koptyuga 4, Novosibirsk, 630090 Russia 2 Novosibirsk State University, ul. Pirogova 2, Novosibirsk, 630090 Russia Received October 11, 2012; in final form, June 6, 2013 Abstract—Using the switching method, we give a classification for the Steiner quadruple systems of order N> 8 and rank r N (different by 2 from the rank of the Hamming code of length N ) which are embedded into the extended perfect binary codes of length N and the same rank. Some lower and upper bounds are provided on the number of these different systems. The lower bound and description of different Steiner quadruple systems of order N and rank r N which are not embedded into the extended perfect binary codes of length N and the same rank are given. DOI: 10.1134/S1990478913040078 Keywords: Steiner quadruple system, extended perfect binary code, switching, il- and ijkl- components, rank INTRODUCTION Let F n be the n-dimensional metric space over the Galois field GF (2) with respect to the Hamming metric. The Hamming distance d(x, y) between every pair of vectors x and y from F n is the number of coordinates in which x and y differ. The Hamming weight w(x) of x ∈ F n is the number of nonzero coordinates of x. A nonempty subset C of F n is a binary code. A vector subspace of F n is a binary linear code. The elements of C are called codewords. The parameters of a binary code C from F n are denoted by (n, |C |,d), where n is the length of the codewords (elements of the code), |C | is the size of the code, and d is the minimum distance of the code (i.e., the minimum Hamming distance between the codewords). The set of nonzero coordinate entries of a vector x ∈ F n is called a support of x and denoted by supp(x). A binary code C of length n with distance d =2d ′ +1 is called perfect one-error correcting (further mentioned as perfect) if, for every vector x ∈ F n , there exists only one codeword y in C such that d(x, y)=1. A linear perfect code of length n, called the Hamming code (we denote it by H n ), is unique up to equivalence. It is known ([10]) that perfect codes have the following parameters: length n =2 r − 1 with r> 1, 2 n−r codewords, and the minimum distance 3. Let C be the extended perfect code of length N =2 r obtained from a perfect code C of length 2 r − 1, r ≥ 2, by parity checking; i.e., adding the coordinate entry equals the sum by modulo 2 of all other entries. In the sequel, we will consider only perfect and extended perfect codes containing all-zero vector. The rank of a code C is the dimension of the linear span of C in F n . It is said that the code C ′ =(C \M ) ∪ M ′ is obtained by a switching of M to M ′ in the binary code C if C ′ has the same parameters as C , see [1]. The set M is called a component of C . The set M is called the il-component of the code C of length N obtained from C by extending by lth coordinate if M ′ = M ⊕ e i ⊕ e l for some i ∈{1, 2,...,N }, where e i and e l are the vectors of weight 1 with 1 in the ith and lth coordinate entries respectively. The set R is called the ijkl-component of C if R is the t 1 t 2 -component for every t 1 ,t 2 ∈{i, j, k, l}. * E-mail: daryik@rambler.ru ** E-mail: sol@math.nsc.ru 522