Journal of Combinatorial Designs最新文献

In the present paper, we give an Assmus–Mattson type theorem for near-extremal Type I and even formally self-dual codes. We show the existence of 1-designs or 2-designs for these codes. As a corollary, we prove the uniqueness of a self-orthogonal 2-$��\left(�\; ��16�\; �,�\; �6�\; �,�\; �8��\; �\right)�$ design.

This work shows several direct and recursive constructions of ordered covering arrays (OCAs) using projection, fusion, column augmentation, derivation, concatenation, and Cartesian product. Upper bounds on covering codes in Niederreiter–Rosenbloom–Tsfasman (shorten by NRT) spaces are also obtained by improving a general upper bound. We explore the connection between ordered covering arrays and covering codes in NRT spaces, which generalize similar results for the Hamming metric. Combining the new upper bounds for covering codes in NRT spaces and ordered covering arrays, we improve upper bounds on covering codes in NRT spaces for larger alphabets. We give tables comparing the new upper bounds for covering codes to existing ones.

This paper is a further contribution to the classification of point-primitive finite regular linear spaces. Let $��S�$ be a nontrivial finite regular linear space whose number of points $��v�$ is squarefree. We prove that if $��G�\le �\text{Aut}��\left(�S�\right)��$ is point-primitive with an alternating socle, then $��S�$ is the projective space $��\text{PG}��\left(��3�,�2��\right)��$.

In this paper we investigate the intersections of classes of covering arrays (CAs) and packing arrays (PAs). The arrays appearing in these intersections obey to upper and lower bounds regarding the appearance of tuples in sub-matrices—we call these arrays balanced covering arrays. We formulate and formalize first observations for which upper and lower bounds on the appearance of tuples it is of interest to consider these intersections of CAs and PAs. Outside of these bounds the intersections will be either empty, for the case of too restrictive constraints, or equal to the maximum element in the emerging lattices, for the case of too weak constraints. We present a column extension algorithm for classification of nonequivalent balanced CAs that uses a SAT solver or a pseudo-Boolean (PB) solver to compute the columns suitable for array extension together with a lex-leader ordering to identify unique representatives for each equivalence class of balanced CAs. These computations bring to light a dissection of classes of CAs that is partially nested due to the nature of the considered intersections. These dissections can be trivial, containing only a single type of balanced CAs, or can also appear as highly structured containing multiple nested types of balanced CAs. Our results indicate that balanced CAs are an interesting class of designs that is rich of structure.

We show that the maximum number of columns in $���E��\left(�s^2�\right)��$-optimal supersaturated designs (SSDs) with 16 rows and $��s_{\max }�=�4�$ is 60 by showing that there exists no resolvable 2-(16, 8, 35) design such that any two blocks from different parallel classes intersect in 3, 5, or 4 points. This is accomplished by an exhaustive computer search that uses the parallel class intersection pattern method to reduce the search space. We also classify all nonisomorphic $���E��\left(�s^2�\right)��$-optimal 5-circulant SSDs with 16 rows and $��s_{\max }�=�8�$.

Coherent configurations are a generalization of association schemes. Motivated by the recent study of Q-polynomial coherent configurations, in this paper, we study the spherical embedding of a Q-polynomial coherent configuration into some eigenspace by a primitive idempotent. We present a necessary and sufficient condition when the embedding becomes a Euclidean $��t�$-design (on two concentric spheres) in terms of the Krein numbers for $��t�\; �\le �\; �4�$. In addition, we obtain some Euclidean 2- or 3-designs from spherical embedding of coherent configurations including tight relative 4- or 5-designs in binary Hamming schemes and the union of derived designs of a tight 4-design in Hamming schemes.