Quadruple Systems Of Order Research Articles

On Affine-Invariant Two-Fold Quadruple Systems

In this paper, we present a direct construction and a recursive construction of the affine-invariant 2-fold quadruple systems of order q (3-(q, 4, 2) designs), where q is a prime power. The direct construction also gives a criterion for the existence of affine-invariant 2-fold quadruple systems of prime order by using 1-factors of graphs. By our recursive construction, an infinite family of affine-invariant 2-fold quadruple systems is established by using our previous direct construction.

Steiner quadruple systems of small rank and extended perfect binary codes

Using the switching method, we give a classification for the Steiner quadruple systems of orderN > 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.

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

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.

Classification of Steiner quadruple systems of order 16 and rank at most 131

A Steiner quadruple system SQS(v) of order v is a 3-design T (v, 4, 3, ?) with ? = 1. In this paper we describe all nonisomorphic systems SQS(16) that can be obtained by the generalized concatenated construction (GC-construction). These Steiner systems have rank at most 13 over $$ \mathbb{F} $$ 2. In particular, there is one system SQS(16) of rank 11 (points and planes of the a fine geometry AG(4, 2)), fifteen systems of rank 12, and 4131 systems of rank 13. All these Steiner systems are resolvable.

Classification of steiner quadruple systems of order 16 and rank at most 13

Classification of steiner quadruple systems of order 16 and rank at most 13