Understanding the maximum size of a code with a given minimum distance is a major question in computer science and discrete mathematics. The most fruitful approach for finding asymptotic bounds on such codes is by using Delsarte's theory of association schemes. With this approach, Delsarte constructs a linear program such that its maximum value is an upper bound on the maximum size of a code with a given minimum distance. Bounding this value can be done by finding solutions to the corresponding dual linear program. Delsarte's theory is very general and goes way beyond binary codes. In this work, we provide universal bounds in the framework of association schemes that generalize the Elias-Bassalygo bound, which can be applied to any association scheme constructed from a distance function. These bounds are obtained by constructing new solutions to Delsarte's dual linear program. We instantiate these results and we recover known bounds for $q$-ary codes and for constant-weight binary codes. Our other contribution is to recover, for essentially any $Q$-polynomial scheme, MRRW-type solutions to Delsarte's dual linear program which are inspired by the Laplacian approach of Friedman and Tillich instead of using the Christoffel-Darboux formulas. We show in particular how the second linear programming bound can be interpreted in this framework.

翻译：理解给定最小距离下码的最大规模是计算机科学和离散数学中的一个核心问题。寻找此类码渐近界的最有效方法是利用Delsarte的结合方案理论。通过该方法，Delsarte构建了一个线性规划，其最大值即为给定最小距离下码最大规模的上界。通过求解相应的对偶线性规划可获得该上界的界。Delsarte理论具有高度普适性，其应用远不止于二元码。本工作在结合方案框架下提出了推广Elias-Bassalygo界的通用界，该界可应用于任何基于距离函数构建的结合方案。这些界是通过构造Delsarte对偶线性规划的新解而获得的。我们将这些结果具体化，恢复了$q$元码和定重二元码的已知界。另一项贡献是：对几乎所有$Q$多项式方案，我们恢复了Delsarte对偶线性规划的MRRW型解——这些解受Friedman和Tillich的拉普拉斯方法启发，而非使用Christoffel-Darboux公式。我们特别展示了如何在此框架下解释第二线性规划界。