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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 structures of matrices, new classes of such polytopes are obtained from known small polytopes and give ML decodable codes by an LP method. This concept &#x201C;consolidation&#x201D; is applied to find a new compact graph which is known as an approach for the graph isomorphism problem. The minimum distances associated with Kendall tau and Euclidean distances of a code obtained by changing the basis of a permutation code may be larger than the original one.

On ML-certificate linear constraints for rank modulation with linear programming decoding and its application to compact graphs