Steiner Quadruple System Research Articles

Semi-Boolean Steiner quadruple Systems and dimensional dual hyperovals

A dimensional dual hyperoval satisfying property (H) (6) in a projective space of order 2 is naturally associated with a ''semi-Boolean'' Steiner quadruple system. The only known examples are associated with Boolean systems. For every d > 2, we construct a new d- dimensional dual hyperoval satisfying property (H) in PGðdðd þ 3Þ=2; 2Þ; its related semi- Boolean system is the Teirlinck one. It is universal and admits quotients in PGðn; 2Þ, with 4d < n < dðd þ 3Þ=2, if d d 6. We also prove the uniqueness of d-dimensional dual hyperovals satisfying property (H) in PGðdðd þ 3Þ=2; 2Þ, whose related semi-Boolean systems belongs to a particular class, which includes Boolean and Teirlinck systems. Finally, we prove property (mI) (6) for them.

A lower bound on the number of Semi‐Boolean quadruple systems

AbstractA Steiner quadruple system of order 2n is Semi‐Boolean (SBQS(2n) in short) if all its derived triple systems are isomorphic to the point‐line design associated with the projective geometry PG(n−1, 2). We prove by means of explicit constructions that for any n, up to isomorphism, there exist at least 2⌊ 3(n−4)/2⌋ regular and resolvable SBQS(2n). © 2003 Wiley Periodicals, Inc. J Combin Designs 11: 229–239, 2003; Published online in Wiley InterScience (www.interscience.wiley.com). DOI 10.1002/jcd.10050

An improved product construction of rotational Steiner quadruple systems

AbstractAn improved product construction is presented for rotational Steiner quadruple systems. Direct constructions are also provided for small orders. It is known that the existence of a rotational Steiner quadruple system of order υ+1 implies the existence of an optimal optical orthogonal code of length υ with weight four and index two. New infinite families of orders are also obtained for both rotational Steiner quadruple systems and optimal optical orthogonal codes. © 2002 Wiley Periodicals, Inc. J Combin Designs 10: 433–443, 2002; Published online in Wiley InterScience (www.interscience.wiley.com). DOI 10.1002/jcd.10025

Piotrowski's infinite series of Steiner Quadruple Systems revisited

The construction of Bays and deWeck [1] of a Steiner Quadruple System SQS(14) was generalized by Piotrowski in his dissertation ([7], p. 34) to an SQS(2p), p \equiv 7 \mod 12 with a group transitive on the points. However he gave no proof of his construction and his presesntation was open to misinterpretation. So Hanfried Lenz suggested to analyse Piotrowski's construction and to supply it with a proof. In the following we will present Piotrowski's ideas somewhat differently and will furnish a proof of the construction.

Colouring steiner quadruple systems

A Steiner quadruple system of order ν (briefly SQS(ν)) is a pair ( X, B ), where | X| = ν and B is a collection of 4-subsets of X, called blocks, such that each 3-subset of X is contained in a unique block of B . A SQS(ν) exists iff ν ≡ 2, 4 (mod 6) or ν = 0, 1 (the admissible integers). The chromatic number of ( X, B ) is the smallest m for which there is a map ϕ: X → Z m such that |ϕ(β)| ⩾ 2 for all β ϵ B . In this paper it is shown that for each m ⩾ 6 there exists ν m such that for all admissible ν ⩾ ν m there exists an m-chromatic SQS(ν). For m = 4, 5 the same statement is proved for admissible ν with the restriction that ν ≢ 2 (mod 12).

Blocking Sets in SQS(2v)

A Steiner quadruple system SQS(v) of order v is a family ℬ of 4-element subsets of a v-element set V such that each 3-element subset of V is contained in precisely one B ∈ ℬ. We prove that if T ∩ B ≠ ø for all B ∈ ℬ (i.e., if T is a transversal), then |T| ≥ v/2, and if T is a transversal of cardinality exactly v/2, then V \ T is a transversal as well (i.e., T is a blocking set). Also, in respect of the so-called ‘doubling construction’ that produces SQS(2v) from two copies of SQS(v), we give a necessary and sufficient condition for this operation to yield a Steiner quadruple system with blocking sets.

Codes of Steiner triple and quadruple systems

The code over a finite fieldF q of orderq of a design is the subspace spanned by the incidence vectors of the blocks. It is shown here that if the design is a Steiner triple system on ν points, and if the integerd is such that 2 d −1≤ν<2 d+1−1, then the binary code of the design contains a subcode that can be shortened to the binary Hamming codeH d of length 2 d −1. Similarly the binary code of any Steiner quadruple system on ν+1 points contains a subcode that can be shortened to the Reed-Muller code ℜ(d−2,d) of orderd−2 and length 2 d , whered is as above.

The structure of nilpotent steiner quadruple systems

AbstractSteiner quadruple systems can be coordinatized by SQS‐skeins. We investigate those Steiner quadruple systems that correspond to finite nilpotent SQS‐skeins. S. Klossek has given representation and construction theorems for finite distributive squags and Hall triple systems which were generalized by the author to the class of all finite nilpotent squags and their corresponding Steiner triple systems. In this article we present analogous theorems for nilpotent SQS‐skeins and Steiner quadruple systems. We also generalize the well‐known doubling constructions of Doyen/Vandensavel and Armanious. It is then possible to describe the structure of all nilpotent Steiner quadruple systems completely: the nilpotent Steiner quadruple systems are exactly those obtained from the trivial 2‐ (or 4‐) element Steiner quadruple system by repeated application of this generalized doubling construction. Moreover, we prove that the variety of semi‐boolean SQS‐skeins is not locally finite and contains non‐nilpotent SQS‐skeins. © 1993 John Wiley &amp; Sons, Inc.

Threshold detection of a quadruple pulse signaling in noisy optical channels

AbstractIn references [1, 2], triple multipulse position modulated (TPPM) signaling has been introduced as a very reliable format for optical communication channels. In this article, such a multipulse model is extended to the Steiner quadruple pulse system (SQS). Considering a threshold demodulation scheme for a fully degraded optical channel, performance of the Reed‐Solomon coded model is evaluated and compared to the standard coded PPM, when similar information efficiency and closest bandwidth expansion are allowed. The gained results are very interesting. © 1992 John Wiley &amp; Sons, Inc.

Subdesigns in Steiner quadruple systems

A Steiner quadruple system of order v, denoted SQS( v), is a pair ( X, B ), where X is a set of cardinality v, and B is a set of 4-subsets of C (called blocks), with the property that any 3-subset of X is contained in a unique block. If ( X, B ) is an SQS( v) and ( Y, C) is an SQS( w) with Y ⊆ X and C ⊆ B , we say that ( Y, C) is a subdesign of ( X, B ). Hanani has shown that an SQS( v) exists for all v ≡ 2 or 4 (mod 6) and when v ∈ {0, 1}; such integers v are said to be admissible. A necessary condition for the existence of an SQS( v) with a subdesign of order w is that v = w or v ⩾ 2 w. In this paper we show the existence of an explicitly computable constant k (independent of w) such that for all admissible v and all admissible w with v ⩾ kw there exists an SQS( v) containing a subdesign of order w. We also show that for any sufficiently large w we can take k = 12.54. To establish these results we introduce several new constructions for SQS, and we also consider the subdesign problem for related classes of designs.

Infinite families of strictly cyclic Steiner quadruple systems

Publisher Summary This chapter describes infinite families of strictly cyclic Steiner Quadruple Systems. A Steiner Quadruple System SQS( v ) of order v is a pair ( V, B ) where V is a set with v elements, B a subset of (y) the elements of which are called blocks so that every 3-subset of v ∊N is contained in a unique block.

Intersection problem of Steiner systems S(3,4,2ν)

A Steiner quadruple system of order v (SQS(v)) is a pair (Q, q), where Q is a v-set and q is a collection of 4-element subsets of Q, called blocks, such that every 3-element subset of Q is contained in exactly one block of q. Hanani [12] proved that an SQS(v) exists if and only if v =-2 or 4 (mod 6). It is easy to see that [ql = ~ v ( v 1)(v 2), which we will denote by q~ in what follows. Denote by J[v] the set of all positive integers k such that there exists a pair of SQS(v) which have exactly k blocks in common, and set I [ v ] = {0, 1, 2 , . . . , q~, 14} U {qv 12, q~ 8, qo}. In [6], Gionfriddo and Lindner conjectured that J[v] = l[v] for every v 2 or 4 (mod 6) and v /> 8. Since the conjecture by Giofiiddo and Lindner, a considerable amount of work has been done in an at tempt to prove that J[v] = I[v] [3-11]. But compared to the whole problem, we still have a lot of work to do. In this paper, we proved that J[2v] = l[2v] for certain admissible order v of Steiner quadruple systems, and l [ 2 v ] {q2~ h : h = 17, 18, 19} ~_J[2v] for every v -= 2 or 4 (mod 6). Moreover, with the assumption of J[16] = I[16], J[20] = I[20], and J[28] = I[28], we are able to prove that J[2v] = I[2v], where v 2 or 4 (mod 6) and v 1> 4.

The existence of resolvable Steiner quadruple systems

A Steiner quadruple system of order v is a set X of cardinality v, and a set Q, of 4-subsets of X, called blocks, with the property that every 3-subset of X is contained in a unique block. A Steiner quadruple system is resolvable if Q can be partitioned into parallel classes (partitions of X). A necessary condition for the existence of a resolvable Steiner quadruple system is that v≡4 or 8 (mod 12). In this paper we show that this condition is also sufficient for all values of v, with 24 possible exceptions.

Orthogonal line packings of PG2 m − 1 (2)

Baker ( Discrete Math., 15 (1976), 205–211) has shown that there exists a packing of the lines of each odd dimensional projective space over the field of two elements as a corollary to a theorem asserting the existence of a 2-resolution of the Steiner quadruple system of planes in an even dimensional affine space over the field of two elements. Two packings are orthogonal if any two of their spreads have at most one line in common. A variation of the previous construction gives alternate packings so that, for example, the existence of orthogonal packings of PG 2 m − 1 (2) when three does not divide 2 m − 1 can be demonstrated.

Product constructions for cyclic block designs. I. Steiner quadruple systems

Let CSQS(ν) denote a cyclic Steiner quadruple system of order ν. Various product constructions for CSQS( uν) are given. Amongst other results we show how to construct a CSQS( uν) ( ν ≢ 0 (mod 4)) from a CSQS( u), a CSQS(2 u) and a CSQS( ν).

A general recursive construction for quadruple systems

A Steiner-quadruple system of order υ is an ordered pair ( X, Q), where X is a set of cardinality υ, and Q 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. In this paper we show that if there exists a quadruple system of order V with a subsystem of order υ, then there exists a quadruple system of order 3 V − 2 υ with subsystems of orders V and υ. Hanani has given a proof of this result for υ = 1, and in a previous paper, the author has proved the case when V ≡ 2 υ(mod 6). The construction given here proves all remaining cases, and has many applications to other existence problems for 3-designs.

Colouring steiner quadruple systems

An algorithm is described which finds a proper 2-colouring of a Steiner quadruple system if one exists; the time required by this method is polynomial. By way of contrast, 2-colouring a partial Steiner quadruple system is shown to be NP-complete.

A cyclic steiner quadruple system of order 32

A cyclic steiner quadruple system of order 32

On projective and affine hyperplanes

We study projective and affine factors in 2-coverings and associate canonically a projective space of order 2 and an affine space of order 3 to each Steiner triple system and an affine space of order 2 to each Steiner quadruple system.

Construction of Steiner Quadruple Systems Having Large Numbers of Nonisomorphic Associated Steiner Triple Systems

If $(Q,q)$ is a Steiner quadruple system and $x$ is any element in $Q$ it is well known that the set ${Q_x} = Q\backslash \{ x\}$ equipped with the collection $q(x)$ of all triples $\{ a,b,c\}$ such that $\{ a,b,c,x\} \in q$ is a Steiner triple system. A quadruple system $(Q,q)$ is said to have at least $n$ nonisomorphic associated triple systems (NATS) provided that for at least one subset $X$ of $Q$ containing $n$ elements the triple systems $({Q_x},q(x))$ and $({Q_y},q(y))$ are nonisomorphic whenever $x \ne y \in X$. Prior to the results in this paper the maximum number of known NATS for any quadruple system was 2. The main result in this paper is the construction for each positive integer $t$ of a quadruple system having at least $t$ NATS.