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 Hamming bound and 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 but which didn't come from the linear program method. 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：理解具有给定最小距离的码的最大尺寸是计算机科学与离散数学中的一个重要问题。寻找此类码渐近界的最有效方法是运用Delsarte的结合方案理论。通过该方法，Delsarte构造了一个线性规划，其最大值即为给定最小距离码最大尺寸的上界。求解该值可通过寻找对应其对偶线性规划的解来实现。Delsarte的理论具有高度普适性，远不止应用于二进制码。本文在结合方案框架下提供了普适性界，推广了Hamming界和Elias-Bassalygo界——这些界可应用于任何由距离函数构造的结合方案。这些结果通过构造Delsarte对偶线性规划的新解获得。我们将这些结果具体化，并恢复了$q$元码与常重二进制码的已知界，而这些界此前并非源自线性规划方法。我们的另一贡献是：本质上对任意$Q$-多项式结合方案，恢复了受Friedman与Tillich的拉普拉斯方法启发（而非使用Christoffel-Darboux公式）的MRRW型Delsarte对偶线性规划解。我们特别展示了第二线性规划界如何在该框架下得到解释。