Quantum jump codes are quantum error-correcting codes which correct errors caused by quantum jumps. A t-spontaneous emission error design (t-SEED) was introduced by Beth et al. in 2003 [T. Beth, C. Charnes, M. Grassl, G. Alber, A. Delgado, and M. Mussinger, A new class of designs which protect against quantum jumps, Des Codes Cryptogr 29 (2003), 51–70.] to construct quantum jump codes. The number of designs (dimension) in a t-SEED corresponds to the number of orthogonal basis states in a quantum jump code. A nondegenerate t-SEED is optimal if it has the largest possible dimension. In this paper, we investigate the bounds on the dimensions of 2-SEEDs systematically. The exact dimensions of optimal 2- SEEDs are almost determined, with five possible exceptions in doubt. General upper bounds on dimensions of 2- SEEDs are demonstrated, the corresponding leave graphs are described, and several exceptional cases are studied in details. Meanwhile, we employ 2-homogenous groups to obtain new lower bounds on the dimensions of 2- SEEDs for prime power orders v and general block sizes k.
In this article, we show that if is a nontrivial nonsymmetric design admitting a flag-transitive point-primitive automorphism group G, then G must be an affine or almost simple group. Moreover, if the socle of G is sporadic, then is the unique 2 − (176, 8, 2) design with , the Higman–Sims simple group.
The construction of group divisible designs (GDDs) is a basic problem in design theory. While there have been some methods concerning the constructions of uniform GDDs, the construction of nonuniform GDDs remains a challenging problem. In this paper, we present a new approach to the construction of nonuniform GDDs with group type and block size k. We make a progress by proposing a new construction, in which generalized difference sets, a truncating technique, and a difference method are combined to construct nonuniform GDDs. Moreover, we present a variation of this new construction by employing Rees' product constructions. We obtain several infinite families of nonuniform GDDs, as well as many examples whose block sizes are relatively large.
Difference systems of sets (DSSs) are combinatorial structures arising in connection with code synchronization that were introduced by Levenshtein in 1971, and are a generalization of cyclic difference sets. In this paper, we consider a collection of m-subsets in a finite field of prime order to be a regular DSS for an integer m, and give a lower bound on the parameter ρ of the DSS using cyclotomic numbers. We show that when we choose -subsets from the multiplicative group of order e, the lower bound on ρ is independent of the choice of subsets. In addition, we present some computational results for DSSs with block sizes and , whose parameter ρ attains or comes close to the Levenshtein bound for .
Generalizing a result by Buratti et al.[M. Buratti, F. Rania, and F. Zuanni, Some constructions for cyclic perfect cycle systems, Discrete Math 299 (2005), 33–48], we present a construction for i-perfect k-cycle decompositions of the complete m-partite graph with parts of size k. These decompositions are sharply vertex-transitive under the additive group of with R a suitable ring of order m. The construction works whenever a suitable i-perfect map exists. We show that for determining the set of all triples for which such a map exists, it is crucial to calculate the chromatic numbers of some auxiliary graphs. We completely determine this set except for one special case where is the product of two distinct primes, is even, and . This result allows us to obtain a plethora of new i-perfect k-cycle decompositions of the complete graph of order (mod 2k) with k odd. In particular, if k is a prime, such a decomposition exists for any possible i provided that .
In this paper, we describe some necessary and sufficient conditions for a set of coboundaries to yield a cocyclic Hadamard matrix over the dihedral group . Using this characterization, new classification results for certain cohomology classes of cocycles over are obtained, extending existing exhaustive calculations for cocyclic Hadamard matrices over from order 36 to order 44. We also define some transformations over coboundaries, which preserve orthogonality of -cocycles. These transformations are shown to correspond to Horadam's bundle equivalence operations enriched with duals of cocycles.
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.
]]>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.
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 .
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.
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.
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.