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A generalization of Lee codes

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Abstract

Motivated by a problem in computer architecture we introduce a notion of the perfect distance-dominating set (PDDS) in a graph. PDDSs constitute a generalization of perfect Lee codes, diameter perfect codes, as well as other codes and dominating sets. In this paper we initiate a systematic study of PDDSs. PDDSs related to the application will be constructed and the non-existence of some PDDSs will be shown. In addition, an extension of the long-standing Golomb–Welch conjecture, in terms of PDDS, will be stated. We note that all constructed PDDSs are lattice-like which is a very important feature from the practical point of view as in this case decoding algorithms tend to be much simpler.

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References

  1. Ahlswede R., Blinovsky V.: Lectures on Advances in Combinatorics. Springer, Berlin (2008)

    Book  MATH  Google Scholar 

  2. Ahlswede R., Audinian H.K., Khachatrian L.H.: On perfect codes and related concepts. Des. Codes Cryptogr. 22, 221–237 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  3. AlBdaiwi B.F., Bose B.: Quasi-perfect Lee distance codes. IEEE Trans. Inf. Theory 49, 1535–1539 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  4. AlBdaiwi B.F., Livingston M.L.: Perfect distance d-placements in 2d-toroidal networks. J. Supercomput. 29, 45–57 (2004)

    Article  MATH  Google Scholar 

  5. Araujo C., Dejter I.J.: On perfectly dominating \({\mathbb{Z}^{3}}\) with squares (submitted).

  6. Bange D.W., Barkauskas A.E., Slater P.J.: Efficient dominating sets in graphs. In: Ringeisen R.D., Roberts F.S. (eds.) Applications in Discrete Mathematics, pp. 189–199. SIAM, Philadelphia (1988).

  7. Biggs N.: Algebraic Graph Theory. Cambridge University Press, Cambridge (1993)

    Google Scholar 

  8. Buzaglo S., Etzion T.: Tilings by (0.5, n)-Crosses and Perfect Codes. online: arXiv:1107.5706v1.

  9. Costa S.I., Muniz M., Agustini E., Palazzo R.: Graphs, tessellations, and perfect codes on flat tori. IEEE Trans. Inf. Theory 50, 2363–2377 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  10. Dejter I.J.: Perfect domination in regular grid graphs. Australas. J. Comb. 42, 99–114 (2008)

    MATH  MathSciNet  Google Scholar 

  11. Dejter I.J., Delgado A.A.: Perfect domination in rectangular grid graphs. J. Comb. Math. Comb. Comput. 70, 177–196 (2009)

    MATH  MathSciNet  Google Scholar 

  12. Dejter I.J., Phelps K.T.: Ternary Hamming and binary perfect covering codes. In: Barg A., Litsyn S. (eds.) Codes and Association Schemes. DIMACS Series Discrete Mathematics and Theoretical Computer Science, vol. 56, pp. 111–113. American Mathematical Society, Providence (2001).

  13. Dejter I.J., Serra O.: Efficient dominating sets in Cayley graphs. Discret. Appl. Math. 119, 319–328 (2003)

    Article  MathSciNet  Google Scholar 

  14. Dejter I.J., Weichsel P.M.: Twisted perfect dominating subgraphs of hypercubes. Congr. Numerantium 94, 67–78 (1993)

    MATH  MathSciNet  Google Scholar 

  15. Etzion T.: Product constructions for perfect Lee codes. IEEE Trans. Inf. Theory (to appear).

  16. Fellows M.R., Hoover M.N.: Perfect domination. Australas. J. Comb. 3, 141–150 (1991)

    MATH  MathSciNet  Google Scholar 

  17. Golomb S., Welch K.: Perfect codes in the Lee metric and the packing of polyominos. SIAM J. Appl. Math. 18, 302–317 (1970)

    Article  MATH  MathSciNet  Google Scholar 

  18. Hickerson D., Stein S.: Abelian groups and packings by semicrosses. Pac. J. Math. 122, 96–109 (1986)

    Article  MathSciNet  Google Scholar 

  19. Horak P., AlBdaiwi B.F.: Non-periodic tilings of R n by crosses. Discret. Comput. Geom. 47, 1–16 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  20. Horak P., AlBdaiwi B.F.: Diameter perfect Lee codes. IEEE Trans. Inf. Theory. Online: arXiv:1109.3475 (to appear).

  21. Klostermeyer W.F., Goldwasser J.L.: Total perfect codes in grid codes. Bull. Inst. Comb. Appl. 46, 61–68 (2006)

    MATH  MathSciNet  Google Scholar 

  22. Kratochvil J., Krivánek M.: On the computational complexity of codes in graphs. In: Proceedings of the MFCS 1988. Lecture Notes in Computer Science, vol. 324, pp. 396–404, Springer, Berlin.

  23. Minkowksi H.: Dichteste gitterformige Lagerung kongruenter Korper. Nachrichten Ges. Wiss. Gottingen, pp. 311–355 (1904).

  24. Molnár E.: Sui Mosaici dello spazio de dimensione n. Atti della Accademia Nazionale dei Lincei. Rend. Sc. Fis. Mat. e Nat. 51, 177–185 (1971)

    MATH  Google Scholar 

  25. Schwartz M.: Quasi-cross lattice tilings with applications to flash memory (submitted).

  26. Stein S.: Factoring by subsets. Pac. J. Math. 22, 523–541 (1967)

    Article  MATH  Google Scholar 

  27. Stein S.: Packings of R n by certain error spheres. IEEE Trans. Inf. Theory 30, 356–363 (1984)

    Article  MATH  Google Scholar 

  28. Szabó S.: On mosaics consisting of multidimensional crosses. Acta Math. Acad. Sci. Hung. 38, 191–203 (1981)

    Article  MATH  Google Scholar 

  29. Weichsel P.M.: Dominating sets of n-cubes. J. Graph Theory 18, 479–488 (1994)

    Article  MATH  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. University of Puerto Rico, Rio Piedras, PR, 00936-8377, USA

    C. Araujo & I. Dejter

  2. University of Washington, Tacoma, WA, 98402, USA

    P. Horak

Authors
  1. C. Araujo
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  2. I. Dejter
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  3. P. Horak
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Correspondence to I. Dejter.

Additional information

This paper is dedicated to the memory of Lucia Gionfriddo.

This is one of several papers published in Designs, Codes and Cryptography comprising the “Special Issue on Coding Theory and Applications”.

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Araujo, C., Dejter, I. & Horak, P. A generalization of Lee codes. Des. Codes Cryptogr. 70, 77–90 (2014). https://doi.org/10.1007/s10623-012-9666-6

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  • DOI: https://doi.org/10.1007/s10623-012-9666-6

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