A new approach towards the Golomb-Welch conjecture

The Golomb-Welch conjecture deals with the existence of perfect e-error correcting Lee codes of word length n, PL(n,e) codes. Although there are many papers on the topic, the conjecture is still far from being solved. In this paper we initiate the study of an invariant connected to abelian groups that enables us to reformulate the conjecture, and then to prove the non-existence of linear PL(n,2) codes for [email protected]?12. Using this new approach we also construct the first quasi-perfect Lee codes for dimension n=3, and show that, for fixed n, there are only finitely many such codes over Z.

[1]  Simon Špacapan.  Optimal Lee-Type Local Structures in Cartesian Products of Cycles and Paths , 2007, SIAM J. Discret. Math..

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

[3]  A. D. Sands,et al.  Factoring Groups into Subsets , 2009 .

[4]  Elwyn R. Berlekamp,et al.  Algebraic Coding Theory: Revised Edition , 2015 .

[5]  Elwyn R. Berlekamp,et al.  Algebraic coding theory , 1984, McGraw-Hill series in systems science.

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

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

[8]  Peter Horák,et al.  Diameter Perfect Lee Codes , 2012, IEEE Transactions on Information Theory.

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

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

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

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

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

[14]  Paul H. Siegel,et al.  Lee-metric BCH codes and their application to constrained and partial-response channels , 1994, IEEE Trans. Inf. Theory.

[15]  Rudolf Ahlswede,et al.  Non-binary error correcting codes with noiseless feedback, localized errors, or both , 2006, 2006 IEEE International Symposium on Information Theory.

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

[17]  Bella Bose,et al.  Quasi-perfect Lee distance codes , 2003, IEEE Trans. Inf. Theory.

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

[19]  Peter Horák,et al.  Fast decoding of quasi-perfect Lee distance codes , 2006, Des. Codes Cryptogr..

[20]  L. J. Boya,et al.  On Regular Polytopes , 2012, 1210.0601.

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

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

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

[24]  Kevin Keating,et al.  Isohedral Polyomino Tiling of the Plane , 1999, Discret. Comput. Geom..

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