Journal of Combinatorial Designs最新文献

Cameron defined the concept of generalized $��t�$-designs, which generalized $��t�$-designs, resolvable designs and orthogonal arrays. This paper introduces a new class of combinatorial designs which simultaneously provide a generalization of both generalized $��t�$-designs and group divisible $��t�$-designs. In certain cases, we derive necessary conditions for the existence of generalized group divisible $��t�$-designs, and then point out close connections with various well-known classes of designs, including mixed orthogonal arrays, factorizations of the complete multipartite graphs, large sets of group divisible designs, and group divisible designs with (orthogonal) resolvability. Moreover, we investigate constructions for generalized group divisible $��t�$-designs and almost completely determine their existence for $��t�=�2�,�3�$ and small block sizes.

Let $��A�C�G��\left(��2�\nu �,�F_q��\right)��$ be the $��2�\nu �$-dimensional classical affine space with parameter $��e�$ over a $��q�$-element finite field $��F_q�$, and $��O_{\nu }�$ be the set of all maximal totally isotropic flats in $��A�C�G��\left(��2�\nu �,�F_q��\right)��$. In this paper, we discuss Cameron–Liebler sets in $��O_{\nu }�$

In the present paper, we give proofs of the existence of a 3-design in the extended ternary quadratic residue code of length 14 and the extended quaternary quadratic residue code of length 18.

We present the number of totally symmetric quasigroups (equivalently, totally symmetric Latin squares) of order 16, as well as the number of isomorphism classes of such objects. Totally symmetric quasigroups of orders up to and including 16 that are (respectively) medial, idempotent, and unipotent are also enumerated.

Steiner triple systems (STSs) have been classified up to order 19. Earlier estimations of the number of isomorphism classes of STSs of order 21, the smallest open case, are discouraging as for classification, so it is natural to focus on the easier problem of merely counting the isomorphism classes. Computational approaches for counting STSs are here considered and lead to an algorithm that is used to obtain the number of isomorphism classes for order 21: 14,796,207,517,873,771.

A Latin square of order $��n�$ is an $��n�\; �\times �\; �n�$ matrix of $��n�$ symbols, such that each symbol occurs exactly once in each row and column. For an odd prime power $��q�$ let $��F_q�$ denote the finite field of order $��q�$. A quadratic Latin square is a Latin square $��L�\; ��\left[�\; ��a�\; �,�\; �b��\; �\right]��$ defined by

A line coloring of $��\text{PG}�\; ��\left(�\; ��n�\; �,�\; �q��\; �\right)��$, the $��n�$-dimensional projective space over GF$��\left(�\; �q�\; �\right)�$, is an assignment of colors to all lines of $��\text{PG}�\; ��\left(�\; ��n�\; �,�\; �q��\; �\right)��$ so that any two lines with the same color do not intersect. The chromatic index of $��\text{PG}�\; ��\left(�\; ��n�\; �,�\; �q��\; �\right)��$, denoted by