Diameter Perfect Lee Codes

Lee codes have been intensively studied for more than 40 years. Interest in these codes has been triggered by the Golomb-Welch conjecture on the existence of the perfect error-correcting Lee codes. In this paper, we deal with the existence and enumeration of diameter perfect Lee codes. As main results, we determine all q for which there exists a linear diameter-4 perfect Lee code of word length n over Zq, and prove that for each n ≥ 3, there are uncountable many diameter-4 perfect Lee codes of word length n over Z. This is in a strict contrast with perfect error-correcting Lee codes of word length n over Z as there is a unique such code for n=3, and its is conjectured that this is always the case when 2n+1 is a prime. We produce diameter perfect Lee codes by an algebraic construction that is based on a group homomorphism. This will allow us to design an efficient algorithm for their decoding. We hope that this construction will turn out to be useful far beyond the scope of this paper.

[1]  Sueli I. Rodrigues Costa,et al.  Graphs, tessellations, and perfect codes on flat tori , 2004, IEEE Transactions on Information Theory.

[2]  Sherman K. Stein,et al.  Algebra and Tiling by Sherman K. Stein , 2009 .

[3]  Simon Špacapan,et al.  Nonexistence of face-to-face four-dimensional tilings in the Lee metric , 2007, Eur. J. Comb..

[4]  P. Horak.  Tilings in Lee metric , 2009, Eur. J. Comb..

[5]  Peter Horak,et al.  Error-correcting codes and Minkowski’s conjecture , 2010 .

[6]  Italo J. Dejter,et al.  A generalization of Lee codes , 2014, Des. Codes Cryptogr..

[7]  C. Y. Lee,et al.  Some properties of nonbinary error-correcting codes , 1958, IRE Trans. Inf. Theory.

[8]  Moshe Schwartz,et al.  Quasi-Cross Lattice Tilings With Applications to Flash Memory , 2011, IEEE Transactions on Information Theory.

[9]  Lorenzo Milazzo,et al.  Enumerating and decoding perfect linear Lee codes , 2009, Des. Codes Cryptogr..

[10]  Sherman K. Stein.  Factoring by subsets , 1967 .

[11]  Sherman Stein,et al.  Combinatorial packings of R3 by certain error spheres , 1984, IEEE Trans. Inf. Theory.

[12]  H. Minkowski.  Dichteste gitterförmige Lagerung kongruenter Körper , 1904 .

[13]  S. Golomb,et al.  Perfect Codes in the Lee Metric and the Packing of Polyominoes , 1970 .

[14]  Werner Ulrich,et al.  Non-binary error correction codes , 1957 .

[15]  Rudolf Ahlswede,et al.  Lectures on advances in combinatorics , 2008, Universitext.

[16]  Dean Hickerson,et al.  Abelian groups and packing by semicrosses. , 1986 .

[17]  Sherman Stein.  Packings of Rn by certain error spheres , 1984, IEEE Trans. Inf. Theory.

[18]  Karel A. Post.  Nonexistence Theorems on Perfect Lee Codes over Large Alphabets , 1975, Inf. Control..

[19]  Sylvain Gravier,et al.  On the Non-existence of 3-Dimensional Tiling in the Lee Metric , 1998, Eur. J. Comb..

[20]  Ladislaus Rédei.  Neuer Beweis des Hamjósschen Satzes über die endlichen Abelschen Gruppen , 1955 .

[21]  S. Szabó.  On mosaics consisting of multidimensional crosses , 1981 .

[22]  P. Horak.  On perfect Lee codes , 2009, Discret. Math..

[23]  Rudolf Ahlswede,et al.  On Perfect Codes and Related Concepts , 2001, Des. Codes Cryptogr..

[24]  Tuvi Etzion,et al.  Product Constructions for Perfect Lee Codes , 2011, IEEE Transactions on Information Theory.

[25]  Peter Horák,et al.  Non-periodic Tilings of ℝn by Crosses , 2012, Discret. Comput. Geom..