The problem we consider in this article is motivated by data placement, in particular data replication in distributed storage and retrieval systems. We are given a set V of v servers along with b files (data, documents). Each file is replicated on exactly k servers. A placement consists in finding a family of b subsets of V (representing the files) called blocks, each of size k. Each server has some probability to fail and we want to find a placement that minimizes the variance of the number of available files. It was conjectured that there always exists an optimal placement (with variance better than any other placement for any value of the probability of failure). We show that the conjecture is true, if there exists a well-balanced design—that is, a family of blocks—each of size k, such that each j-element subset of V, , belongs to the same or almost the same number of blocks (difference at most one). The existence of well-balanced designs is a difficult problem as it contains, as a subproblem, the existence of Steiner systems. We completely solve the case and give bounds and constructions for and some values of v and b.
In this note, we present a simple doubling construction for 3-uniform friendship hypergraphs which generalizes the cubeconstructed hypergraphs from another study (L. Jørgensen and A. Sillasen, J Combin Designs (2014)). As a by-product, we build point-transitive 3-uniform friendship hypergraphs of sizes and for all .
Abstract: A Kakeya set in the linear representation , a nonsingular conic, is the point set covered by a set of lines, one through each point of . In this article, we classify the small Kakeya sets in . The smallest Kakeya sets have size , and all Kakeya sets with weight less than are classified: there are approximately types.
]]>A triple system is a collection of b blocks, each of size three, on a set of v points. It is j-balanced when every two j-sets of points appear in numbers of blocks that are as nearly equal as possible, and well balanced when it is j-balanced for each . Well-balanced systems arise in the minimization of variance in file availability in distributed file systems. It is shown that when a triple system that is 2-balanced and 3-balanced exists, so does one that is well balanced. Using known and new results on variants of group divisible designs, constructions for well-balanced triple systems are developed. Using these, the spectrum of pairs for which such a well-balanced triple system exists is determined completely.
Let G be a graph of order n satisfying that there exists for which every graph of order n and size t is contained in exactly λ distinct subgraphs of the complete graph isomorphic to G. Then G is called t-edge-balanced and λ the index of G. In this article, new examples of 2-edge-balanced graphs are constructed from bipartite graphs and some further methods are introduced to obtain more from old.
For each odd , we completely solve the problem of when an m-cycle system of order u can be embedded in an m-cycle system of order v, barring a finite number of possible exceptions. In cases where u is large compared to m, where m is a prime power, or where , the problem is completely resolved. In other cases, the only possible exceptions occur when is small compared to m. This result is proved as a consequence of a more general result that gives necessary and sufficient conditions for the existence of an m-cycle decomposition of a complete graph of order v with a hole of size u in the case where and both hold.
Suppose that and . We construct a Latin square of order n with the following properties:
Hence generalizes the square that Sade famously found to complete Norton's enumeration of Latin squares of order 7. In particular, is what is known as a self-switching Latin square and possesses a near-autoparatopism.
Quantum jump codes are quantum codes that correct errors caused by quantum jumps. A spontaneous emission error design (SEED) was introduced by Beth et al. in 2003 to construct quantum jump codes. In this paper, we study the existence of 3-SEEDs from PSL(2, q) or PGL(2, q). By doing this, a large number of 3- SEEDs are derived for prime powers q and all .
In this paper, we determine the necessary and sufficient conditions for the existence of an equitably ℓ-colorable balanced incomplete block design for any positive integer . In particular, we present a method for constructing nontrivial equitably ℓ-colorable BIBDs and prove that these examples are the only nontrivial examples that exist. We also observe that every equitable ℓ-coloring of a BIBD yields both an equalized ℓ-coloring and a proper 2-coloring of the same BIBD.
This paper examines subsets with at most n points on a line in the projective plane . A lower bound for the size of complete -arcs is established and shown to be a generalisation of a classical result by Barlotti. A sufficient condition ensuring that the trisecants to a complete (k, 3)-arc form a blocking set in the dual plane is provided. Finally, combinatorial arguments are used to show that, for , plane (k, 3)-arcs satisfying a prescribed incidence condition do not attain the best known upper bound.
]]>An is a triple , where X is a set of points, is a partition of X into m disjoint sets of size n and is a set of 4-element transverses of , such that each 3-element transverse of is contained in exactly one of them. If the full automorphism group of an admits an automorphism α consisting of n cycles of length m (resp. m cycles of length n), then this is called m-cyclic (resp. semi-cyclic). Further, if all block-orbits of an m-cyclic (resp. semi-cyclic) are full, then it is called strictly cyclic. In this paper, we construct some infinite classes of strictly m-cyclic and semi-cyclic , and use them to give new infinite classes of perfect two-dimensional optical orthogonal codes with maximum collision parameter and AM-OPPTS/AM-OPPW property.
A cube design of order v, or a CUBE(v), is a decomposition of all cyclicly oriented quadruples of a v-set into oriented cubes. A CUBE(v) design is unoriented if its cubes can be paired so that the cubes in each pair are related by reflection through the center. A cube design is degenerate if it has repeated points on one of its cubes, otherwise it is nondegenerate.
We show that a nondegenerate CUBE(v) design exists for all integers , and that an unoriented nondegenerate CUBE(v) design exists if and only if and or . A degenerate example of a CUBE(v) design is also given for each integer .
Using the technique of amalgamation-detachment, we show that the complete equipartite multigraph can be decomposed into cycles of lengths (plus a 1-factor if the degree is odd) whenever there exists a decomposition of into cycles of lengths (plus a 1-factor if the degree is odd). In addition, we give sufficient conditions for the existence of some other, related cycle decompositions of the complete equipartite multigraph .
We introduce the notion of a partial geometric difference family as a variation on the classical difference family and a generalization of partial geometric difference sets. We study the relationship between partial geometric difference families and both partial geometric designs and difference families, and show that partial geometric difference families give rise to partial geometric designs. We construct several infinite families of partial geometric difference families using Galois rings and the cyclotomy of Galois fields. From these partial geometric difference families, we generate a list of infinite families of partial geometric designs and directed strongly regular graphs.
Let L be a latin square of indeterminates. We explore the determinant and permanent of L and show that a number of properties of L can be recovered from the polynomials and per(L). For example, it is possible to tell how many transversals L has from per(L), and the number of 2 × 2 latin subsquares in L can be determined from both and per(L). More generally, we can diagnose from or per(L) the lengths of all symbol cycles. These cycle lengths provide a formula for the coefficient of each monomial in and per(L) that involves only two different indeterminates. Latin squares A and B are trisotopic if B can be obtained from A by permuting rows, permuting columns, permuting symbols, and/or transposing. We show that nontrisotopic latin squares with equal permanents and equal determinants exist for all orders that are divisible by 3. We also show that the permanent, together with knowledge of the identity element, distinguishes Cayley tables of finite groups from each other. A similar result for determinants was already known.
We consider the existence problem for a semi-cyclic holey group divisible design of type with block size 3, which is denoted by a 3-SCHGDD of type . When t is odd and or t is doubly even and , the existence problem is completely solved; when t is singly even, many infinite families are obtained. Applications of our results to two-dimensional balanced sampling plans and optimal two-dimensional optical orthogonal codes are also discussed.
A pseudo-hyperoval of a projective space , q even, is a set of subspaces of dimension such that any three span the whole space. We prove that a pseudo-hyperoval with an irreducible transitive stabilizer is elementary. We then deduce from this result a classification of the thick generalized quadrangles that admit a point-primitive, line-transitive automorphism group with a point-regular abelian normal subgroup. Specifically, we show that is flag-transitive and isomorphic to , where is either the regular hyperoval of PG(2, 4) or the Lunelli–Sce hyperoval of PG(2, 16).
Latin hypercube designs have been found very useful for designing computer experiments. In recent years, several methods of constructing orthogonal Latin hypercube designs have been proposed in the literature. In this article, we report some more results on the construction of orthogonal Latin hypercubes which result in several new designs.
Candelabra quadruple systems (CQS) were first introduced by Hanani who used them to determine the existence of Steiner quadruple systems. In this paper, a new method has been developed by constructing partial candelabra quadruple systems with odd group size, which is a generalization of the even cases, to complete a design. New results of candelabra quadruple systems have been obtained, i.e. we show that for any , there exists a CQS for all , and .
We detail the enumeration of all two-intersection sets of the five-dimensional projective space over the field of order 3 that are invariant under an element of order 7, which include the examples of Hill (1973) and Gulliver (1996). Up to projective equivalence, there are 6,635 such two-intersection sets.
We present a construction for minimal blocking sets with respect to -spaces in , the -dimensional projective space over the finite field of order . The construction relies on the use of blocking cones in the field reduced representation of , extending the well-known construction of linear blocking sets. This construction is inspired by the construction for minimal blocking sets with respect to the hyperplanes by Mazzocca, Polverino, and Storme (the MPS-construction); we show that for a suitable choice of the blocking cone over a planar blocking set, we obtain larger blocking sets than the ones obtained from planar blocking sets in F. Mazzocca and O. Polverino, J Algebraic Combin 24(1) (2006), 61–81. Furthermore, we show that every minimal blocking set with respect to the hyperplanes in can be obtained by applying field reduction to a minimal blocking set with respect to -spaces in . We end by relating these constructions to the linearity conjecture for small minimal blocking sets. We show that if a small minimal blocking set is constructed from the MPS-constructionthen it is of Rédei-type, whereas a small minimal blocking set arises from our cone construction if and only if it is linear.