A pair of orthogonal latin cubes of order q is equivalent to a maximum distance separable code with distance 3 or to an OA1(3,5,q) orthogonal array. We construct pairs of orthogonal latin cubes for ...sequences of previously unknown orders qi=16(18i−1)+4 and qi′=16(18i+5)+4. The minimal new obtained parameters of orthogonal arrays are OA1(3,5,84).
In this paper, we consider the problem of constructing hypercycle systems of 5-cycles in complete 3-uniform hypergraphs. A hypercycle system C(r,k,v) of order v is a collection of r-uniform k-cycles ...on a v-element vertex set, such that each r-element subset is an edge in precisely one of those k-cycles. We present cyclic hypercycle systems C(3,5,v) of orders v=25,26,31,35,37,41,46,47,55,56, a highly symmetric construction for v=40, and cyclic 2-split constructions of orders 32,40,50,52. As a consequence, all orders v≤60 permitted by the divisibility conditions admit a C(3,5,v) system. New recursive constructions are also introduced.
Steiner systems are a fascinating topic of combinatorics. The most studied Steiner systems are S(2,3,v) (Steiner triple systems), S(3,4,v) (Steiner quadruple systems), and S(2,4,v). There are a few ...infinite families of Steiner systems S(2,4,v) in the literature. The objective of this paper is to present an infinite family of Steiner systems S(2,4,2m) for all m≡2(mod4)≥6 from cyclic codes. As a by‐product, many infinite families of 2‐designs are also reported in this paper.
Storage architectures ranging from minimum bandwidth regenerating encoded distributed storage systems to declustered-parity RAIDs can employ dense partial Steiner systems to support fast reads, ...writes, and recovery of failed storage units. To enhance performance, popularities of the data items should be taken into account to make frequencies of accesses to storage units as uniform as possible. A combinatorial model ranks items by popularity and assigns data items to elements in a dense partial Steiner system so that the sums of ranks of the elements in each block are as equal as possible. By developing necessary conditions in terms of independent sets, we demonstrate that certain Steiner systems must have a much larger difference between the largest and smallest block sums than is dictated by an elementary lower bound. In contrast, we also show that certain dense partial
S
(
t
,
t
+
1
,
v
)
designs can be labeled to realize the elementary lower bound. Furthermore, we prove that for every admissible order
v
, there is a Steiner triple system (
S
(2, 3,
v
)) whose largest difference in block sums is within an additive constant of the lower bound.
A (v,3,2)-covering is a family of 3-subsets of a v-set, called blocks, such
that any two elements of v-set appear in at least one of the blocks. In this
paper, we propose new construction of ...(v,3,2)-coverings with the minimum
number of blocks. This construction represents a generalization of Bose?s and
Skolem?s constructions of Steiner systems S(2,3,6n+3) and S(2,3,6n+1). Unlike
the existing constructions, our construction is direct and it uses the set of
base blocks and permutation p, so by applying it the remaining blocks of
(v,3,2)-covering are obtained.
In 1977, D. Betten defined a geometric group to be a permutation group (G,Ω) such that G=Aut(R) for some hypergraph
R on
Ω. In this paper, we extend Betten's notion of a geometric group to what we ...call a geometric group of second order. By definition, this is a permutation group for which
G
=
Aut
(
R
) for some set
R
=
{
R
1
,
R
2
,
…
,
R
d
} of hypergraphs on
Ω. Our main focus will be on permutation groups that are geometric of second order but not geometric. Within this small class of groups one finds the projective groups
P
G
L
(
2
,
8
)
,
P
Γ
L
(
2
,
8
) and the affine groups
A
G
L
(
1
,
8
)
,
A
Γ
L
(
1
,
8
). Our investigations, which are based primarily on these four groups, lead us to consider some familiar combinatorial structures (eg, Fano plane and affine design) in a less familiar context (overlarge sets of Steiner systems).
In Woldar (Geometric groups of second order and related combinatorial structures, J. Combin. Designs 28(4) (2020), 307–326), the second author gave a computer-free construction of three overlarge ...sets of Steiner systems: two of type
S
(
2
,
3
,
7
)
(the Fano plane) and one of type
S
(
3
,
4
,
8
)
(affine design). Presently, we use these constructions to derive four computer-free models of partial geometries. Two of these geometries are of type
pg
(
8
,
9
,
4
)
, one of which is classical. The remaining two, which are of type
pg
(
5
,
7
,
3
)
, are due to Mathon, who also proved that these are the only partial geometries with (5, 7, 3).
We establish that the logarithm of the number of latin d‐cubes of order n is Θ(ndlnn) and the logarithm of the number of sets of t (t≥2 is fixed) orthogonal latin squares of order n is Θ(n2lnn). ...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 Θ(n3lnn) as n→∞ and n mod 6=2 or 4.
Topology‐transparent scheduling (TTS) for wireless networks involves generation of schedules which do not need update even when the network topology is constantly changing. TTS schemes designed with ...Steiner systems support the largest number of nodes—among all cover‐free families—for a given frame length. In this paper, we propose a TTS scheme using Steiner systems for a network in which nodes are divided into multiple classes of different priorities. Our proposed scheme called “Steiner system‐based Topology‐transparent Priority Scheduling” (STPS) produces priority‐sensitive schedules using different Steiner systems for the different priority classes. We explain our proposed method through illustrative examples and show its superiority over a competitive scheme in the literature with respect to the frame length.
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