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Quasi-affine symmetric designs

机译：拟仿射对称设计

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摘要

A symmetric design with parameters v = q~2(q + 2), k = q(q + 1), λ = q, q ≥ 2, is called a quasi-affine design if its point set can be partitioned into q + 2 subsets P_0, P_1, ..., P_q,P_(q+1) such that the induced structure in every point neighborhood is an affine plane of order q (repeated q times). A quasi-affine design with q ≥ 3 determines its point neighborhoods uniquely and dual of such a design is also a quasi-affine design. These structural properties pave way for definition of a strongly quasi-affine design and it is also shown that associated with every quasi-affine design is a unique strongly quasi-affine design from which the given quasi-affine design is obtained by certain unique cutting and pasting operation. This investigation also enables us to associate a unique 2-regular graph with q + 2 vertices and in turn, a unique colored partition of the integer q + 2. These combinatorial consequences are finally used to obtain an exponential lower bound on the number of non-isomorphic solutions of such symmetric designs improving the earlier lower bound of 2.
机译：参数v = q〜2（q + 2），k = q（q + 1），λ= q，q≥2的对称设计，如果其点集可以划分为q +，则称为准仿射设计2个子集P_0，P_1，...，P_q，P_（q + 1），以使每个点邻域中的诱导结构都是q阶的仿射平面（重复q次）。 q≥3的拟仿射设计唯一确定其点邻域，并且这种设计的对偶也是拟仿射设计。这些结构特性为定义强拟仿射设计铺平了道路，并且还表明与每个拟仿射设计相关联的是独特的强拟仿射设计，可以通过某些独特的切割方法从中获得给定的拟仿射设计。粘贴操作。此研究还使我们能够将唯一的2正则图与q + 2顶点关联，进而将整数q + 2的唯一彩色分区关联。这些组合结果最终用于获得非n的数量的指数下界。对称设计的同构解提高了2的较早下限。

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