Steiner Quadruple System Research Articles

Completely Regular Codes in Johnson and Grassmann Graphs with Small Covering Radii

Let ${\cal L}$ be a Desarguesian 2-spread in the Grassmann graph $J_q(n,2)$. We prove that the collection of the $4$-subspaces, which do not contain subspaces from ${\cal L}$ is a completely regular code in $J_q(n,4)$. Similarly, we construct a completely regular code in the Johnson graph $J(n,6)$ from the Steiner quadruple system of the extended Hamming code. We obtain several new completely regular codes with covering radius $1$ in the Grassmann graph $J_2(6,3)$ using binary linear programming.

New infinite classes of 2‐chromatic Steiner quadruple systems

AbstractIn 1971, Doyen and Vandensavel gave a special doubling construction that gives a direct construction of 2‐chromatic Steiner quadruple system of order (SQS) for all or . The first author presented a construction for 2‐chromatic SQSs based on 2‐chromatic candelabra quadruple systems and ‐fan designs. In this paper, it is proved that a 2‐chromatic SQS also exists if , or if .

Symmetric abelian group-invariant Steiner quadruple systems

Let K be an abelian group of order v. A Steiner quadruple system of order v (SQS(v)) (K,B) is called symmetric K-invariant if for each B∈B, it holds that B+x∈B for each x∈K and B=−B+y for some y∈K. When the Sylow 2-subgroup of K is cyclic, a necessary and sufficient condition for the existence of a symmetric K-invariant SQS(v) was given by Munemasa and Sawa, which is a generalization of a necessary and sufficient condition for the existence of a symmetric cyclic SQS(v) shown in Piotrowski's thesis in 1985. In this paper, we prove that a symmetric K-invariant SQS(v) exists if and only if v≡2,4(mod6), the order of each element of K is not divisible by 8, and there exists a symmetric cyclic SQS(2p) for any odd prime divisor p of v.

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.

Existence of Steiner quadruple systems with an almost spanning block design

A Steiner quadruple system of order v (SQS(v)) is said to be have an almost spanning block design and denoted by 1-AFSQS(v) if it contains a subdesign S(2,4,v−1). In 1992, Hartman and Phelps posed a problem: Show that there exists a 1-AFSQS(v) for each v≡2(mod12). In this paper, we prove that the necessary condition for the existence of a 1-AFSQS(v) is also sufficient with a definite exception v=14 and possible exceptions v∈{86,206,374,398}.

New optical orthogonal signature pattern codes with maximum collision parameter 2 and weight 4

Optical orthogonal signature pattern codes (OOSPCs) play an important role in a novel type of optical code-division multiple-access network for 2-dimensional image transmission. There is a one-to-one correspondence between an $$(m, n, w, \lambda )$$ -OOSPC and a $$(\lambda +1)$$ -(mn, w, 1) packing design admitting an automorphism group isomorphic to $$\mathbb {Z}_m\times \mathbb {Z}_n$$ . In 2010, Sawa gave a construction of an (m, n, 4, 2)-OOSPC from a one-factor of Kohler graph of $$\mathbb {Z}_m\times \mathbb {Z}_n$$ which contains a unique element of order 2. In this paper, we study the existence of one-factor of Kohler graph of $$\mathbb {Z}_m\times \mathbb {Z}_n$$ having three elements of order 2. It is proved that there is a one-factor in the Kohler graph of $$\mathbb {Z}_{2^{\epsilon }p}\times \mathbb {Z}_{2^{\epsilon '}}$$ relative to the Sylow 2-subgroup if there is an S-cyclic Steiner quadruple system of order 2p, where $$p\equiv 5\pmod {12}$$ is a prime and $$1\le \epsilon ,\epsilon '\le 2$$ . Using this one-factor, we construct a strictly $$\mathbb {Z}_{2^{\epsilon }p}\times \mathbb {Z}_{2^{\epsilon '}}$$ -invariant regular $$G^*(p,2^{\epsilon +\epsilon '},4,3)$$ relative to the Sylow 2-subgroup. By using the known S-cyclic SQS(2p) and a recursive construction for strictly $$\mathbb {Z}_{m}\times \mathbb {Z}_{n}$$ -invariant regular G-designs, we construct more strictly $$\mathbb {Z}_{m}\times \mathbb {Z}_{n}$$ -invariant 3-(mn, 4, 1) packing designs. Consequently, there is an optimal $$(2^{\epsilon }m,2^{\epsilon '}n,4,2)$$ -OOSPC for any $$\epsilon ,\epsilon '\in \{0,1,2\}$$ with $$\epsilon +\epsilon '>0$$ and an optimal (6m, 6n, 4, 2)-OOSPC where m, n are odd integers whose all prime divisors from the set $$\{p\equiv 5\pmod {12}:p$$ is a prime, $$p<$$ 1,500,000}.

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 last twenty orders of [formula omitted]-resolvable Steiner quadruple systems

A Steiner quadruple system (X,B) is said to be (1,2)-resolvable if its blocks can be partitioned into r parts such that each point of X occurs in exactly two blocks in each part. The necessary condition for the existence of (1,2)-resolvable Steiner quadruple systems RSQS(1,2,v)s is v≡2 or 10 (mod 12). 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] posed a question whether the necessary condition for the existence of (1,2)-resolvable Steiner quadruple systems is sufficient. In this paper, we consider the last twenty orders of (1,2)-resolvable Steiner quadruple systems and show that the necessary condition for the existence of (1,2)-resolvable Steiner quadruple systems is also sufficient except for the order 10.

A new existence proof for Steiner quadruple systems

A Steiner quadruple system of order v is an ordered pair \({(X, \mathcal{B})}\), where X is a set of cardinality v, and \({\mathcal{B}}\) is a set of 4-subsets of X, called blocks, with the property that every 3-subset of X is contained in a unique block. Such designs exist if and only if \({v \equiv 2,4\, (\bmod\, 6)}\). The first and second proofs of this result were given by Hanani in 1960 and in 1963, respectively. All the existing proofs are rather cumbersome, even though simplified proofs have been given by Lenz in 1985 and by Hartman in 1994. To study Steiner quadruple systems, Hanani introduced the concept of H-designs in 1963. The purpose of this paper is to provide an alternative existence proof for Steiner quadruple systems via H-designs of type 2 n . In 1990, Mills showed that for n > 3, n ≠ 5, an H-design of type g n exists if and only if ng is even and g(n − 1)(n − 2) is divisible by 3, where the main context is the complicated existence proof for H-designs of type 2 n . However, Mill’s proof was based on the existence result of Steiner quadruple systems. In this paper, by using the theory of candelabra systems and H-frames, we give a new existence proof for H-designs of type 2 n independent of the existence result of Steiner quadruple systems. As a consequence, we also provide a new existence proof for Steiner quadruple systems.

Existence of Steiner quadruple systems with a spanning block design

A Steiner system S ( t , k , v ) is a pair ( X , B ) , where X is a v -element set and B is a set of k -subsets of X , called blocks, with the property that every t -element subset of X is contained in a unique block. The sub-design S ( 2 , 4 , v ) in a Steiner quadruple system S ( 3 , 4 , v ) is said to be a spanning block design. The order v of a Steiner quadruple system with a spanning block design should satisfy the necessary condition v ≡ 4 ( mod 12 ) . It is proved that the above necessary condition is also sufficient. As a consequence, it is also proved that a 3-BD S ( 3 , { 4 , 5 } , v ) exists for any v ≡ 5 ( mod 12 ) .

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.

H-designs with the properties of resolvability or (1, 2)-resolvability

An H-design is said to be (1, α)-resolvable, if its block set can be partitioned into ?-parallel classes, each of which contains every point of the design exactly α times. When α = 1, a (1, α)-resolvable H-design of type gn is simply called a resolvable H-design and denoted by RH(gn), for which the general existence problem has been determined leaving mainly the case of g ≡ 0 (mod 12) open. When α = 2, a (1, 2)-RH(1n) is usually called a (1, 2)-resolvable Steiner quadruple system of order n, for which the existence problem is far from complete. In this paper, we consider these two outstanding problems. First, we prove that an RH(12n) exists for all n ≥ 4 with a small number of possible exceptions. Next, we give a near complete solution to the existence problem of (1, 2)-resolvable H-designs with group size 2. As a consequence, we obtain a near complete solution to the above two open problems.

Almost simple groups with socle [formula omitted] acting on Steiner quadruple systems

Let N = L n ( q ) , n ⩾ 2 , q a prime power, be a projective linear simple group. We classify all Steiner quadruple systems admitting a group G with N ⩽ G ⩽ Aut ( N ) . In particular, we show that G cannot act as a group of automorphisms on any Steiner quadruple system for n > 2 .

On orthogonal resolutions of the classical Steiner quadruple system SQS(16)

A Steiner quadruple system SQS(16) is a pair $${(V, \mathcal{B})}$$ where V is a 16-set of objects and $${\mathcal B}$$ is a collection of 4-subsets of V, called blocks, so that every 3-subset of V is contained in exactly one block. By classical is meant the boolean quadruple system, also known as the affine geometry AG(4,2). A parallel class is a collection of four blocks which partition V. The system possesses a resolution or parallelism, since $${\mathcal B}$$ can be partitioned into 35 parallel classes. Two resolutions are called orthogonal when each parallel class of one resolution has at most one block in common with each parallel class of the other resolution. We prove that there are at most nine further resolutions which, together with the classical one, are pairwise orthogonal.

Small Group Divisible Steiner Quadruple Systems

A group divisible Steiner quadruple system, is a triple $(X, {\cal H}, {\cal B})$ where $X$ is a $v$-element set of points, ${\cal H} = \{H_1,H_2,\ldots,H_r\}$ is a partition of $X$ into holes and ${\cal B}$ is a collection of $4$-element subsets of $X$ called blocks such that every $3$-element subset is either in a block or a hole but not both. In this article we investigate the existence and non-existence of these designs. We settle all parameter situations on at most 24 points, with 6 exceptions. A uniform group divisible Steiner quadruple system is a system in which all the holes have equal size. These were called by Mills G-designs and their existence is completely settled in this article.

Some constructions for block sequences of Steiner quadruple systems with error correcting consecutive unions

AbstractIn this article, we investigate a block sequence of a Steiner quadruple system which contains the blocks exactly once such that the collection of all blocks together with all unions of two consecutive blocks of the sequence forms an error correcting code with minimum distance four. In particular, we give two recursive constructions and obtain infinitely many such sequences by utilizing individual sequences as starters of the recursions. © 2007 Wiley Periodicals, Inc. J Combin Designs 16: 152–163, 2008

The spectrum of equitable, 4‐chromatic Steiner quadruple systems

AbstractA Steiner quadruple system of order v (briefly an SQS(v)) is a pair (X,$\cal B$) with |X| = v and $\cal B$ a set of quadruples taken from X such that every triple in X is in a unique quadruple in $\cal B$. Hanani [Canad J Math 12 (1960), 145–157] showed that an SQS(v) exists if and only if v is {admissible}, that is, v = 0,1 or v ≡ 2,4 (mod 6). Each SQS(v) has a chromatic number when considered as a 4‐uniform hypergraph. Here we show that a 4‐chromatic SQS(v) exists for all admissible v ≥ 20, and that no 4‐chromatic SQS(v) exists for v &lt; 20. Each system we construct admits a proper 4‐coloring that is equitable, that is, any two color classes differ in size by at most one. © 2006 Wiley Periodicals, Inc. J Combin Designs 15: 369–392, 2007

Existence of 3‐chromatic Steiner quadruple systems

AbstractA Steiner quadruple system of order v (briefly SQS (v)) is a pair (X, $\cal B$), where X is a v‐element set and $\cal B$ is a set of 4‐element subsets of X (called blocks or quadruples), such that each 3‐element subset of X is contained in a unique block of $\cal B$. The chromatic number of an SQS(v)(X, $\cal B$) is the smallest m for which there is a map $\varphi : X \rightarrow Z_m$ such that $|\varphi(B)|\geq 2$ for all $B \in \cal B$, where $\varphi (B) =\{\varphi (x):x\in B\}$. The system (X, $\cal B$) is equitably m‐chromatic if there is a proper coloring $\varphi$ with minimal m for which the numbers $|\varphi^{-1}(c)|, c\in Z_m$ differ from each other by at most 1. Linek and Mendelsohn showed that an equitably 3‐chromatic SQS(v) exists for v ≡ 4, 8, 10 (mod 12), v ≥ 16. In this article we show that an equitably 3‐chromatic SQS(v) exists for v ≡ 2 (mod 12) with v &gt; 2. © 2006 Wiley Periodicals, Inc. J Combin Designs 15: 469–477, 2007

Resolvable Steiner Quadruple Systems for the Last 23 Orders

A Steiner quadruple system of order v, denoted by SQS(v), is a pair $(X,{\cal B}),$ where X is a v-set of points and ${\cal B}$ is a set of 4-subsets of X, called blocks, with the property that every 3-subset of X is contained in exactly one block of ${\cal B}$. An SQS(v) is resolvable if its block set can be partitioned into parallel classes. In 1987, Hartman showed that the necessary condition $v\equiv 4$ or $8 (\rm {mod} 12)$ for the existence of a resolvable SQS(v) is sufficient with 23 possible exceptions. In this paper, we construct resolvable SQS(v)s for the 23 undecided orders.

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.