Journal of Combinatorial Designs最新文献

In this paper, we consider the existence of group divisible designs (GDDs) with block size $��4�$ and group sizes $��4�$ and $��7�$. We show that there exists a $��4�$-GDD of type $��4^t�7^s�$ for all but a finite specified set of feasible values for $���\left(��t�,�s��\right)��$.

The main purpose of this paper is to further study the structure, parameters and constructions of the recently introduced minimal codes in the sum-rank metric. These objects form a bridge between the classical minimal codes in the Hamming metric, the subject of intense research over the past three decades partly because of their cryptographic properties, and the more recent rank-metric minimal codes. We prove some bounds on their parameters, existence results, and, via a tool that we name geometric dual, we manage to construct minimal codes with few weights. A generalization of the celebrated Ashikhmin–Barg condition is proved and used to ensure the minimality of certain constructions.

Maximum rank-distance (MRD) codes are (not necessarily linear) maximum codes in the rank-distance metric space on $��m�$-by-$��n�$ matrices over a finite field $��F_q�$. They are diameter perfect and have the cardinality $��q^{�m��\left(��n�-�d�+�1��\right)��}�$ if $��m�\ge �n�$. We define switching in MRD codes as the replacement of special MRD subcodes by other subcodes with the same parameters. We consider constructions of MRD codes admitting switching, such as punctured twisted Gabidulin codes and direct-product codes. Using switching, we construct a huge class of MRD codes whose cardinality grows doubly exponentially in $��m�$ if the other parameters ($��n�,��q�$, the code distance) are fixed. Moreover, we construct MRD codes with different affine ranks and aperiodic MRD codes.

Recently, new combinatorial structures called disjoint partial difference families (DPDFs) and external partial difference families (EPDFs) were introduced, which simultaneously generalize partial difference sets, disjoint difference families and external difference families, and have applications in information security. So far, all known construction methods have used cyclotomy in finite fields. We present the first noncyclotomic infinite families of DPDFs which are also EPDFs, in structures other than finite fields (in particular cyclic groups and nonabelian groups). As well as direct constructions, we present an approach to constructing DPDFs/EPDFs using relative difference sets (RDSs); as part of this, we demonstrate how the well-known RDS result of Bose extends to a very natural construction for DPDFs and EPDFs.

We describe a class of combinatorial design problems which typically occur in professional sailing league competitions. We discuss connections to resolvable block designs and equitable coverings and to scheduling problems in operations research. We in particular give suitable boolean quadratic and integer linear optimization problem formulations, as well as further heuristics and restrictions, that can be used to solve sailing league problems in practice. We apply those techniques to three case studies obtained from real sailing leagues and compare the results with previously used tournament plans.

In this article, we study symmetric designs admitting flag-transitive, point-imprimitive almost simple automorphism groups with socle sporadic simple groups. As a corollary, we present a classification of symmetric designs admitting flag-transitive automorphism group whose socle is a sporadic simple group, and in conclusion, there are exactly seven such designs, one of which admits a point-imprimitive automorphism group and the remaining are point-primitive.