Paratopism is a well-known action of the wreath product on Latin squares of order n. A paratopism that maps a Latin square to itself is an autoparatopism of that Latin square. Let Par(n) denote the set of paratopisms that are an autoparatopism of at least one Latin square of order n. We prove a number of general properties of autoparatopisms. Applying these results, we determine Par(n) for . We also study the proportion of all paratopisms that are in Par(n) as .
An E–W matrix M is a ( − 1, 1)-matrix of order , where t is a positive integer, satisfying that the absolute value of its determinant attains Ehlich–Wojtas' bound. M is said to be of skew type (or simply skew) if is skew-symmetric where I is the identity matrix. In this paper, we draw a parallel between skew E–W matrices and skew Hadamard matrices concerning a question about the maximal determinant. As a consequence, a problem posted on Cameron's website [7] has been partially solved. Finally, codes constructed from skew E–W matrices are presented. A necessary and sufficient condition for these codes to be self-dual is given, and examples are provided for lengths up to 52.
In this paper, we show that partial geometric designs can be constructed from certain three-class association schemes and ternary linear codes with dual distance three. In particular, we obtain a family of partial geometric designs from the three-class association schemes introduced by Kageyama, Saha, and Das in their article [“Reduction of the number of associate classes of hypercubic association schemes,” Ann Inst Statist Math 30 (1978)]. We also give a list of directed strongly regular graphs arising from the partial geometric designs obtained in this paper.
A construction of q-covering designs in PG(5, q) is given, providing an improvement on the upper bound of the q-covering number .
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 .
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.
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 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.
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.