Designs, Codes and Cryptography

, Volume 85, Issue 3, pp 483–521 | Cite as

Consolidation for compact constraints and Kendall tau LP decodable permutation codes



Invented in the 1960s, permutation codes have reemerged in recent years as a topic of great interest because of properties making them attractive for certain modern technological applications, especially flash memory. In 2011 a polynomial time algorithm called linear programming (LP) decoding was introduced for a class of permutation codes where the feasible set of codewords was a subset of the vertex set of a code polytope. In this paper we investigate a new class of linear constraints for matrix polytopes with no fractional vertices through a new concept called “consolidation.” We then introduce a necessary and sufficient condition for which LP decoding methods, originally designed for the Euclidean metric, may be extended to provide an efficient decoding algorithm for permutation codes with the Kendall tau metric.


Coding and information theory Graphs and abstract algebra Linear programming Permutation groups Decoding 

Mathematics Subject Classification

68P30 05C25 90C05 20B 94B35 


  1. 1.
    Andrews G.E.: The Theory of Partitions. Addison-Wesley, Reading (1976).MATHGoogle Scholar
  2. 2.
    Barg A., Mazumdar A.: Codes in permutations and error correction for rank modulation. In: Proceedings of the IEEE ISIT, pp. 854–858 (2010).Google Scholar
  3. 3.
    Berger T., Jelinek F., Wolf J.: Permutation codes for sources. IEEE Trans. Inf. Theory I8, 160–169 (1972).MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Birkhoff G.: Three observations on linear algebra. Univ. Nac. Tacuman Rev. Ser. A 5, 147–151 (1946).MathSciNetMATHGoogle Scholar
  5. 5.
    Blake I.F., Cohen G., Deza M.: Coding with permutations. Inf. Control 43, 1–19 (1979).MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Buchheim C., Cameron P.J., Wu T.: On the subgroup distance problem. Discret. Math. 309, 962–968 (2009).MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Deza M., Huang T.: Metrics on permutations, a survey. JCISS (1998).Google Scholar
  8. 8.
    Fultion W., Harris J.: Representation Theory: A First Course. Springer, New York (1991).Google Scholar
  9. 9.
    Garey M., Johnson D.: Computers and Intractability: A Guide to the Theory of NP-Completeness. W. H. Freeman, New York. ISBN 978-0-7167-1045-5, OCLC 11745039 (1979).Google Scholar
  10. 10.
    Hagiwara M.: On ML-certificate linear constraints for rank modulation with linear programming decoding and its application to compact graphs. In: Proceedings of the IEEE ISIT, pp. 2993–2997 (2012).Google Scholar
  11. 11.
    Hill R.: A First Course in Coding Theory. Oxford Applied Mathematics and Computing Science Series. Oxford University Press, Oxford (1986).Google Scholar
  12. 12.
    Humphreys J.E.: Reflection Groups and Coxeter Groups. Cambridge University Press, Cambridge (1992).MATHGoogle Scholar
  13. 13.
    Jiang A., Schwartz M., Bruck J.: Error-correcting codes for rank modulation. In: Proceedings of the IEEE ISIT, pp. 1736–1740 (2008).Google Scholar
  14. 14.
    Jiang A., Mateescu R., Schwartz M., Bruck J.: Rank modulation for flash memories. IEEE Trans. Inf. Theory 55, 2659–2673 (2009).MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Kane R.: Reflection Groups and Invariant Theory. Springer, New York (2001).CrossRefMATHGoogle Scholar
  16. 16.
    Karmarkar N.: A new polynomial time algorithm for linear programming. Combinatorica 4, 373–395 (1984).MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    Kong J., Hagiwara M.: Comparing Euclidean, Kendall tau metrics toward extending LP decoding. In: Proceedings of the ISITA, pp. 91–95 (2012).Google Scholar
  18. 18.
    Knuth D.: The Art of Computer Programming, vol. 3. Addison-Wesley, Reading (1998).MATHGoogle Scholar
  19. 19.
    Mazumdar A., Barg A., Zemor G.: Constructions of rank modulation codes. IEEE Trans. Inf. Theory 59, 1018–1029 (2012).MathSciNetCrossRefMATHGoogle Scholar
  20. 20.
    Papandreou N., Pozidis H., Mittelholzer T., Close G.F., Breitwisch M., Lam C., Eleftheriou E.: Drift-tolerant multilevel phase-change memory. In: Proceedings of the IEEE IMW, pp. 1–4 (2011).Google Scholar
  21. 21.
    Peterson W.W., Nation J.B., Fossorier M.P.: Reflection group codes and their decoding. IEEE Trans. Inf. Theory 56, 6273–6293 (2010).MathSciNetCrossRefMATHGoogle Scholar
  22. 22.
    Schreck H., Tinhofer G.: A note on certain subpolytopes of the assignment polytope associated with circulant graphs. Linear Algebra Appl. 111, 125–134 (1988).MathSciNetCrossRefMATHGoogle Scholar
  23. 23.
    Slepian D.: Permutation modulation. Proc. IEEE 53, 228–236 (1965).CrossRefGoogle Scholar
  24. 24.
    Tinhofer G.: Graph isomorphism and theorems of Birkhoff type. Computing 36(4), 285–300 (1986).MathSciNetCrossRefMATHGoogle Scholar
  25. 25.
    Tinhofer G.: A note on compact graphs. Discret. Appl. Math. 30(2–3), 253–264 (1991).MathSciNetCrossRefMATHGoogle Scholar
  26. 26.
    Vinck A.J.H.: Coded modulation for powerline communications. AEÜ Int. J. Electron. Commun. 54, 45–49 (2009).Google Scholar
  27. 27.
    Von Neumann J.: A certain zero-sum two-person game equivalent to an optimal assignment problem. Ann. Math. Stud. 28, 5–12 (1953).MathSciNetGoogle Scholar
  28. 28.
    Wadayama T., Hagiwara M.: LP decodable permutation codes based on linearly constrained permutation matrices. In: Proceedings of the IEEE ISIT, pp. 139–143 (2011).Google Scholar

Copyright information

© Springer Science+Business Media New York 2016

Authors and Affiliations

  1. 1.Department of Mathematics and InformaticsChiba UniversityChiba CityJapan

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