Steiner Quadruple Systems Of Order Research Articles

Maximum sets of mutually disjoint 2-chromatic Steiner quadruple systems

This paper concentrates on mutually disjoint 2-chromatic Steiner quadruple systems of order v (briefly by SQS(v)s), meaning that the SQS(v)s are mutually disjoint and they are mutually 2-chromatic having the same two color classes. Following Phelps and Rosa's work, we study maximum sets of mutually disjoint 2-chromatic SQS(v)s by applying direct constructions for suitable collections of triple representatives of Vq by utilizing automorphism groups of affine transformations in the finite fields Fq for prime powers q. Our main result presents many new maximum sets of mutually disjoint 2-chromatic SQS(v)s. We also improve the lower bounds on the number of mutually disjoint SQS(v)s as a by-product.

The number of the non-full-rank Steiner quadruple systems [formula omitted

The p-rank of a Steiner quadruple system B is the dimension of the linear span of the set of characteristic vectors of blocks of B over GF(p). We derive a formula for the number of different Steiner quadruple systems of order v and given 2-rank r, r<v. Our result extends previous work on enumerating Steiner quadruple systems according to the rank of their codes over GF(2), mainly by V.A. Zinoviev and D.V. Zinoviev.

On the number of SQSs, latin hypercubes and MDS codes

AbstractWe establish that the logarithm of the number of latin d‐cubes of order n is and the logarithm of the number of sets of t ( is fixed) orthogonal latin squares of order n is . Similar estimations are obtained for systems of mutually strongly orthogonal latin d‐cubes. As a consequence, we construct a set of Steiner quadruple systems of order n such that the logarithm of its cardinality is as and .

Overlap Cycles for Steiner Quadruple Systems

AbstractSteiner quadruple systems are set systems in which every triple is contained in a unique quadruple. It is well known that Steiner quadruple systems of order v, or SQS(v), exist if and only if . Universal cycles, introduced by Chung, Diaconis, and Graham in 1992, are a type of cyclic Gray code. Overlap cycles are generalizations of universal cycles that were introduced in 2010 by Godbole, et al. Using Hanani's SQS constructions, we show that for every with there exists an SQS(v) that admits a 1‐overlap cycle.

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.

The existence of augmented resolvable Steiner quadruple systems

An augmented Steiner quadruple system of order v is an ordered triple ( X , B , E ) , where ( X , B ) is an SQS ( v ) and E is the set of all 2-subsets of X . An augmented Steiner quadruple system ( X , B , E ) of order v is resolvable if B ∪ E can be partitioned into n = ( v − 1 ) ( v + 4 ) / 6 parts B ∪ E = P 1 | P 2 | ⋯ | P n such that each part P i is a partition of X . Hartman and Phelps in [A. Hartman, K.T. Phelps, Steiner quadruple systems, in: J.H. Dinitz, D.R. Stinson (Eds.), Contemporary Design Theory, Wiley, New York, 1992, pp. 205–240] conjectured that there exists a resolvable augmented Steiner quadruple systems of order v for any positive integer v ≡ 2 or 10 (mod 12). In this paper, we show that the Hartman and Phelps conjecture is true.

Uniformly resolvable three-wise balanced designs with block sizes four and six

A t -wise balanced design is said to be resolvable if its block set can be partitioned into parts (called resolution classes) such that each part is itself a partition of the point set. It is uniform if all blocks in each resolution class have the same size. In this paper, it is shown that a uniformly resolvable three-wise balanced design of order v with block sizes four and six exists if and only if v is divisible by 4. These uniformly resolvable three-wise balanced designs are also used to construct the infinite classes of resolvable maximal packings (minimal coverings) of triples by quadruples of order v for v ≡ 0 ( mod 24 ) , augmented resolvable Steiner quadruple systems of order v for v ≡ 26 , 58 , 74 ( mod 96 ) and ( 1 , 2 ) -resolvable Steiner quadruple systems of order v for v ≡ 74 ( mod 96 ) .

Constructions for rotational Steiner quadruple systems

AbstractA direct construction for rotational Steiner quadruple systems of order p+ 1 having a nontrivial multiplier automorphism is presented, where p≡13 (mod24) is a prime. We also give two improved product constructions. By these constructions, the known existence results of rotational Steiner quadruple systems are extended. © 2009 Wiley Periodicals, Inc. J Combin Designs 17: 353–368, 2009

Classification of steiner quadruple systems of order 16 and rank 14

All 708 103 nonisomorphic Steiner systems S(16, 4, 3) of order 16 and rank 14 over F 2 are enumerated. Among them there are exactly 1059 homogeneous systems. It is shown that all the 708 103 Steiner systems can be obtained by the general doubling construction presented in the paper.

There exist Steiner triple systems of order 15 that do not occur in a perfect binary one‐error‐correcting code

AbstractThe codewords at distance three from a particular codeword of a perfect binary one‐error‐correcting code (of length 2m−1) form a Steiner triple system. It is a longstanding open problem whether every Steiner triple system of order 2m−1 occurs in a perfect code. It turns out that this is not the case; relying on a classification of the Steiner quadruple systems of order 16 it is shown that the unique anti‐Pasch Steiner triple system of order 15 provides a counterexample. © 2006 Wiley Periodicals, Inc. J Combin Designs 15: 465–468, 2007

The Steiner quadruple systems of order 16

The Steiner quadruple systems of order 16 are classified up to isomorphism by means of an exhaustive computer search. The number of isomorphism classes of such designs is 1 , 054 , 163 . Properties of the designs—including the orders of the automorphism groups and the structures of the derived Steiner triple systems of order 15—are tabulated. A consistency check based on double counting is carried out to gain confidence in the correctness of the classification.

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

A class of 2-chromatic SQS(22)

We enumerate all 2-chromatic Steiner quadruple systems of order 22, having a cyclic automorphism of order 11 (i.e. 2 disjoint 11-cycles).

Towards a Large Set of Steiner Quadruple Systems

Let $D( v )$ be the number of pairwise disjoint Steiner quadruple systems. A simple counting argument shows that $D( v )\leqq v - 3$. In this paper it is proved that $D( 2^k n )\geqq ( 2^k - 1 ) n,k\geqq 2$, if there exists a set of $3n$ pairwise disjoint Steiner quadruple systems of order $4n$ with a certain structure. This implies that $D( v )\geqq v - o( v )$ for infinitely many values of v. New lower bounds on $D( v )$ for many values of v that are not divisible by 4 are also given, and it is proved that $D( v )\geqq 2$ for all $v \equiv 2$ or $4(\bmod 6 ),v\geqq 8$.

Construction of Factorized Steiner Quadruple Systems of Order 16 with Blocking Sets

Construction of Factorized Steiner Quadruple Systems of Order 16 with Blocking Sets

Cyclic large sets of Steiner triple systems of order 15

We examine a class of large sets of Steiner triple systems of order 15 having an automorphism consisting of two fixed points and a 13-cycle. We exhibit all members of this class: there are 256 nonisomorphic systems. We examined these members for initial configurations which could lead to a large set of Steiner quadruple systems of order 16 and established that no large set exists having three fixed points and a 13-cycle.

On the set J( v) for Steiner Quadruple systems of order v = 2 n with n⩾4

In the area of the Block-Intersection problem for Steiner Quadruple Systems (see [4, 5]), we prove that q 16−37 = 103 and q 16−29 = 111 ϵ J (16), and that q v − h ϵ J( v) for h = 21, 25, v = 2 n and n⩾4.

Construction of steiner quadruple systems having a prescribed number of blocks in common

Let q υ=υ(υ–1)(υ–2)/24 and let I υ={0, 1, 2, …, q υ–14}∪{q υ –12, q υ–8, q υ}, for υ⩾8 Further, let J[υ] denote the set of all k such that there exists a pair of Steiner quadruple systems of order υ having exactly k blocks in common. We determine J[υ] for all υ=2 n , n⩾2, with the possible exception of 7 cases for υ=16 and of 5 cases for each υ⩾32. In particular we show: J[υ]⊆ I υ for all υ≡2 or 4 (mod 6) and υ⩾8, J[4]={1}, J[8]= I 8={0, 2, 6, 14}, I 16⧹{103, 111, 115, 119, 121, 122, 123}⊆ J[16], and I υ⧹ {q υ– h: h=17, 18, 19, 21, 25}⊆ J[υ] for all υ=2 n , n⩾5.