On ML-Certificate Linear Constraints for Rank Modulation with Linear Programming Decoding and its Application to Compact Graphs

TL;DR

This paper explores linear constraints on matrix polytopes for permutation codes and rank modulation, introducing a consolidation method to create compact graphs and improve code distances, with applications to graph isomorphism.

Contribution

It introduces the concept of consolidation to generate new polytopes from small ones, leading to a compact graph approach for graph isomorphism and enhanced permutation codes.

Findings

New classes of matrix polytopes with no fractional vertices

A novel compact graph for graph isomorphism

Permutation codes with improved minimum distances

Abstract

Linear constraints for a matrix polytope with no fractional vertex are investigated as intersecting research among permutation codes, rank modulations, and linear programming methods. By focusing the discussion to the block structure of matrices, new classes of such polytopes are obtained from known small polytopes. This concept, called "consolidation", is applied to find a new compact graph which is known as an approach for the graph isomorphism problem. Encoding and decoding algorithms for our new permutation codes are obtained from existing algorithms for small polytopes. The minimum distances associated with Kendall-tau distance and the minimum Euclidean distance of a code obtained by changing the basis of a permutation code may be larger than the original one.