Let PG$(\mathbb{F}_q^v)$ be the $(v-1)$-dimensional projective space over $\mathbb{F}_q$ and let $\Gamma$ be a simple graph of order ${q^k-1\over q-1}$ for some $k$. A 2$-(v,\Gamma,\lambda)$ design over $\mathbb{F}_q$ is a collection $\cal B$ of graphs (blocks) isomorphic to $\Gamma$ with the following properties: the vertex-set of every block is a subspace of PG$(\mathbb{F}_q^v)$; every two distinct points of PG$(\mathbb{F}_q^v)$ are adjacent in exactly $\lambda$ blocks. This new definition covers, in particular, the well known concept of a 2$-(v,k,\lambda)$ design over $\mathbb{F}_q$ corresponding to the case that $\Gamma$ is complete. In this pioneer work we illustrate how difference methods allow to get concrete non-trivial examples of $\Gamma$-decompositions over $\mathbb{F}_2$ or $\mathbb{F}_3$ with $\Gamma$ a cycle, a path, a prism, a generalized Petersen graph, or a Moebius ladder. In particular, we will discuss in detail the special and very hard case that $\Gamma$ is complete and $\lambda=1$, i.e., the Steiner 2-designs over a finite field. This study also leads to some (probably new) collateral problems concerning difference sets.

中文翻译：

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射影几何中的图分解

设 PG$(\mathbb{F}_q^v)$ 是 $\mathbb{F}_q$ 上的 $(v-1)$ 维射影空间，让 $\Gamma$ 是阶 ${ 的简单图q^k-1\over q-1}$ 一些 $k$。$\mathbb{F}_q$ 上的 2$-(v,\Gamma,\lambda)$ 设计是与 $\Gamma$ 同构的图（块）的集合 $\cal B$，具有以下属性： -每个块的集合是PG$(\mathbb{F}_q^v)$的一个子空间；PG$(\mathbb{F}_q^v)$ 的每两个不同点在恰好 $\lambda$ 块中相邻。这个新定义特别涵盖了众所周知的概念，即在 $\mathbb{F}_q$ 上的 2$-(v,k,\lambda)$ 设计，对应于 $\Gamma$ 是完整的情况。在这项开创性的工作中，我们说明了差分方法如何允许在 $\mathbb{F}_2$ 或 $\mathbb{F}_3$ 上以 $\Gamma$ 为循环获得 $\Gamma$ 分解的具体非平凡示例，一条路径，一个棱镜，广义彼得森图，或莫比乌斯阶梯。特别地，我们将详细讨论$\Gamma$ 完备且$\lambda=1$ 的特殊和非常困难的情况，即有限域上的Steiner 2-设计。这项研究还导致了一些关于差异集的（可能是新的）附带问题。