AI 中文总结
该研究将组合设计相交数推广至正则半格设计，推导了广义Singleton界等结果，在多经典方案中恢复或得到新分布，为相关设计与编码提供统一处理方法。
AI 中文摘要
我们将组合设计的相交数推广至满足合适正则性条件的有限交半格中的设计。虽然正则半格中的设计可追溯至德尔萨特（Delsarte），但我们的正则性假设比他的更弱，且无需导出结合方案。在该框架下，我们推广了门德尔松（Mendelsohn）方程，证明了以施泰纳（Steiner）系为等号情形的广义辛格尔顿（Singleton）界，并确定了施泰纳系中任意一个区组的区组相交分布；特别地，该分布与所选取的区组无关。将结果专门应用于若干经典半格族时，我们的结果恢复了编码与设计理论中诸多熟知的分布：在汉明（Hamming）和 q-汉明（或称双线性型）方案中，它们分别给出了最大距离可分（MDS）码和最大秩距离（MRD）码的局部距离分布；在约翰逊（Johnson）和 q-约翰逊（或称格拉斯曼（Graßmann））方案中，它们重现了经典及 q-模拟施泰纳系的区组相交分布，等价于直径完美的常重码和直径完美的常维子空间码的距离分布。据我们所知，对于 q-约翰逊方案而言，该结果是新的。作为进一步的说明，我们将我们的理论应用于完美匹配的设计。我们的方法以文献中已知的最强形式对这些情形提供了统一处理，确定了相对于每个单独区组或码字的分布，无需取平均，也无需线性性或可加性假设。此外，它识别出了这些分布所基于的自然双重计数对象，从而得到了以半格的正则性参数表示的公式，避免了通过环境结合方案基于特征值的方法所产生的更为繁琐的表达式。
英文摘要
We generalize intersection numbers for combinatorial designs to designs in finite meet-semilattices satisfying suitable regularity conditions. While designs in regular semilattices go back to Delsarte, our regularity assumptions are weaker than his and need not give rise to an association scheme. In this framework, we extend Mendelsohn's equations, prove a generalized Singleton bound with Steiner systems as equality cases, and determine the block intersection distribution at any block of a Steiner system. In particular, this distribution is independent of the chosen block. Specializing to several classical semilattice families, our results recover a number of well-known distributions in coding and design theory. In the Hamming and the $q$-Hamming (or bilinear forms) schemes, they give the local distance distributions of MDS and MRD codes, respectively. In the Johnson and $q$-Johnson (or Graßmann) schemes, they reproduce the block intersection distribution of classical and $q$-analog Steiner systems, equivalently the distance distribution of diameter-perfect constant-weight codes and diameter-perfect constant-dimension subspace codes. For the $q$-Johnson schemes, to the best of our knowledge, this result is new. As a further illustration, we apply our theory to designs of perfect matchings. Our approach provides a unified treatment of these cases in the strongest form known in the literature, determining the distribution relative to each individual block or codeword, without averaging and without linearity or additivity assumptions. Moreover, it identifies the natural double-counting objects underlying these distributions, leading to formulas in the regularity parameters of the semilattice and avoiding the more cumbersome expressions that arise in eigenvalue-based approaches via the ambient association scheme.
