与Kendall Tau距离度量相关的相干矩阵的置换码长度的线性规划上界
Linear Programming Upper Bounds on Permutation Code Sizes From Coherent Configurations Related to the Kendall Tau Distance Metric
摘要：
最近对排列秩调制的兴趣表明Kendalltau度量是一个重要的距离度量。这篇文章记录了我们第一次努力获得最佳代码大小的上限（对于所述度量）ala Delsarte的方法。对于汉明度量，Delsarte在强大的线性规划（LP）界上的开创性工作已经通过关联方案理论扩展到置换码。对于Kendall tau度量，相同的扩展需要更一般的相干配置理论，从而最佳代码大小问题可以被公式化为一个非常巨大的半定规划（SDP）问题。受最近的代数技术解决SDP的启发，我们考虑了对偶问题，并提出了一个LP搜索的一个子集的对偶可行解。我们获得适度的改进，最近的单例约束由于巴格和Mazumdar。我们把这项工作作为一个起点，充分利用电力Delsarte的方法，这是众所周知的，给一些最好的界限的情况下，二进制代码。
英文摘要：
Recent interest on permutation rank modulation shows the Kendall tau metric as an important distance metric. This note documents our first efforts to obtain upper bounds on optimal code sizes (for said metric) ala Delsarte's approach. For the Hamming metric, Delsarte's seminal work on powerful linear programming (LP) bounds have been extended to permutation codes, via association scheme theory. For the Kendall tau metric, the same extension needs the more general theory of coherent configurations, whereby the optimal code size problem can be formulated as an extremely huge semidefinite programming (SDP) problem. Inspired by recent algebraic techniques for solving SDP's, we consider the dual problem, and propose an LP to search over a subset of dual feasible solutions. We obtain modest improvement over a recent Singleton bound due to Barg and Mazumdar. We regard this work as a starting point, towards fully exploiting the power of Delsarte's method, which are known to give some of the best bounds in the context of binary codes.
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