Designs, Codes and Cryptography

, Volume 75, Issue 1, pp 81–95 | Cite as

Perfect codes in the discrete simplex

Article

Abstract

We study the problem of existence of (nontrivial) perfect codes in the discrete \( n \)-simplex \( \Delta _{\ell }^n := \left\{ \left( \begin{array}{l} x_0, \ldots , x_n \end{array}\right) : x_i \in {\mathbb {Z}}_{+}, \sum _i x_i = \ell \right\} \) under \( \ell _1 \) metric. The problem is motivated by the so-called multiset codes, which have recently been introduced by the authors as appropriate constructs for error correction in the permutation channels. It is shown that \( e \)-perfect codes in the 1-simplex \( \Delta _{\ell }^1 \) exist for any \( \ell \ge 2e + 1 \), the 2-simplex \( \Delta _{\ell }^2\) admits an \( e \)-perfect code if and only if \( \ell = 3e + 1 \), while there are no perfect codes in higher-dimensional simplices. In other words, perfect multiset codes exist only over binary and ternary alphabets.

Keywords

Multiset codes Permutation channel Discrete simplex Perfect codes Sphere packing Integer codes  Manhattan metric 

Mathematics Subject Classification

94B25 05B40 52C17 05C12 68R99 

References

  1. 1.
    Aigner M.: Combinatorial Theory. Springer, New York (1979).Google Scholar
  2. 2.
    AlBdaiwi B., Horak P., Milazzo L.: Enumerating and decoding perfect linear Lee codes. Des. Codes Cryptogr. 52(2), 155–162 (2009).Google Scholar
  3. 3.
    Astola J.: On perfect Lee codes over small alphabets of odd cardinality. Discret. Appl. Math. 4, 227–228 (1982).Google Scholar
  4. 4.
    Bange D.W., Barkauskas A.E., Slater P.J.: Efficient dominating sets in graphs. In: Ringeisen R.D., Roberts F.S. (eds.) Applications of Discrete Mathematics, pp. 189–199. SIAM, Philadelphia (1988).Google Scholar
  5. 5.
    Bertsekas D.P., Gallager R.: Data Networks, 2nd edn. Prentice Hall, Englewood Cliffs (1992).Google Scholar
  6. 6.
    Best M.R.: Perfect codes hardly exist. IEEE Trans. Inf. Theory 29(3), 349–351 (1983).Google Scholar
  7. 7.
    Biggs N.: Perfect codes in graphs. J. Comb. Theory B 15(3), 289–296 (1973).Google Scholar
  8. 8.
    Bours P.A.H.: On the construction of perfect deletion-correcting codes using design theory. Des. Codes Cryptogr. 6(1), 5–20 (1995).Google Scholar
  9. 9.
    Chihara L.: On the zeros of the Askey–Wilson polynomials, with applications to coding theory. SIAM J. Math. Anal. 18(1), 191–207 (1987).Google Scholar
  10. 10.
    Cohen G., Honkala I., Litsyn S., Lobstein A.: Covering Codes. Elsevier, Amsterdam (1997).Google Scholar
  11. 11.
    Delsarte P.: An algebraic approach to association schemes and coding theory. Philips J. Res. 10, 1–97 (1973).Google Scholar
  12. 12.
    Etzion T.: On the nonexistence of perfect codes in the Johnson scheme. SIAM J. Discret. Math. 9(2), 201–209 (1996).Google Scholar
  13. 13.
    Etzion T.: Configuration distribution and designs of codes in the Johnson scheme. J. Comb. Des. 15(1), 15–34 (2007).Google Scholar
  14. 14.
    Etzion T.: Product constructions for perfect Lee codes. IEEE Trans. Inf. Theory 57(11), 7473–7481 (2011).Google Scholar
  15. 15.
    Etzion T., Schwartz M.: Perfect constant-weight codes. IEEE Trans. Inf. Theory 50(9), 2156–2165 (2004).Google Scholar
  16. 16.
    Etzion T., Vardy A.: Perfect binary codes: constructions, properties, and enumeration. IEEE Trans. Inf. Theory 40(3), 754–763 (1994).Google Scholar
  17. 17.
    Etzion T., Vardy A.: Error-correcting codes in projective space. IEEE Trans. Inf. Theory 57(2), 1165–1173 (2011).Google Scholar
  18. 18.
    Gadouleau M., Goupil A.: Binary codes for packet error and packet loss correction in store and forward. In: Proceedings of the International ITG Conference on Source and Channel Coding, Siegen, Germany (2010)Google Scholar
  19. 19.
    Gadouleau M., Goupil A.: A matroid framework for noncoherent random network communications. IEEE Trans. Inf. Theory 57(2), 1031–1045 (2011).Google Scholar
  20. 20.
    Golomb S.W., Welch L.R.: Perfect codes in the Lee metric and the packing of polyominoes. SIAM J. Appl. Math. 18(2), 302–317 (1970).Google Scholar
  21. 21.
    Gordon D.M.: Perfect single error-correcting codes in the Johnson scheme. IEEE Trans. Inf. Theory 52(10), 4670–4672 (2006).Google Scholar
  22. 22.
    Horak P.: Tilings in Lee metric. Eur. J. Comb. 30(2), 480–489 (2009).Google Scholar
  23. 23.
    Horak P.: On perfect Lee codes. Discret. Math. 309(18), 5551–5561 (2009).Google Scholar
  24. 24.
    Kötter R., Kschischang F.R.: Coding for errors and erasures in random network coding. IEEE Trans. Inf. Theory 54(8), 3579–3591 (2008).Google Scholar
  25. 25.
    Kovačević M., Vukobratović D.: Subset codes for packet networks. IEEE Commun. Lett. 17(4), 729–732 (2013).Google Scholar
  26. 26.
    Kovačević M., Vukobratović D.: Multiset codes for permutation channels. Available online at: arXiv:1301.7564.Google Scholar
  27. 27.
    Levenshtein V.I.: On perfect codes in deletion and insertion metric. Discret. Math. Appl. 2(3), 241–258 (1992).Google Scholar
  28. 28.
    MacWilliams F.J., Sloane N.J.A.: The Theory of Error-Correcting Codes. North-Holland, Amsterdam (1977).Google Scholar
  29. 29.
    Martin W.J., Zhu X.J.: Anticodes for the Grassmann and bilinear forms graphs. Des. Codes Cryptogr. 6(1), 73–79 (1995).Google Scholar
  30. 30.
    Post K.A.: Nonexistence theorem on perfect Lee codes over large alphabets. Inf. Control 29(4), 369–380 (1975).Google Scholar
  31. 31.
    Roos C.: A note on the existence of perfect constant weight codes. Discret. Math. 47, 121–123 (1983).Google Scholar
  32. 32.
    Shimabukuro O.: On the nonexistence of perfect codes in \( J(2w + p2, w)\). Ars Comb. 75, 129–134 (2005).Google Scholar
  33. 33.
    Špacapan S.: Non-existence of face-to-face four dimensional tiling in the Lee metric. Eur. J. Comb. 28(1), 127–133 (2007).Google Scholar
  34. 34.
    Tietäväinen A.: On the nonexistence of perfect codes over finite fields. SIAM J. Appl. Math. 24(1), 88–96 (1973).Google Scholar
  35. 35.
    van Lint J. H.: Nonexistence theorems for perfect error-correcting codes. In: Computers in Algebra and Number Theory, vol. IV, SIAM-AMS Proceedings (1971).Google Scholar
  36. 36.
    Zinoviev V.A., Leontiev V.K.: The nonexistence of perfect codes over Galois fields. Probl. Control Inf. Theory 2, 123–132 (1973).Google Scholar

Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  1. 1.Department of Electrical EngineeringUniversity of Novi SadNovi SadSerbia

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