Linear Programming Upper Bounds on Permutation Code Sizes From Coherent Configurations Related to the Kendall Tau Distance Metric

TL;DR

This paper develops linear programming bounds for permutation codes under the Kendall tau metric using coherent configurations, providing initial improvements over existing bounds and paving the way for more powerful code size estimates.

Contribution

It extends Delsarte's LP approach to permutation codes with the Kendall tau metric via coherent configurations, introducing a dual LP formulation for better bounds.

Findings

Achieved modest improvement over the Singleton bound by Barg and Mazumdar.

Formulated a large SDP problem for permutation codes under Kendall tau metric.

Proposed an LP approach to explore dual feasible solutions for better bounds.

Abstract

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.