50 Years of the Golomb–Welch Conjecture

Since 1968, when the Golomb–Welch conjecture was raised, it has become the main motive power behind the progress in the area of the perfect Lee codes. Although, there is a vast literature on the topic and it is widely believed to be true, this conjecture is far from being solved. In this paper, we provide a survey of papers on the Golomb–Welch conjecture. Furthermore, new results on Golomb–Welch conjecture dealing with perfect Lee codes of large radii are presented. Algebraic ways of tackling the conjecture in the future are discussed as well. Finally, a brief survey of research inspired by the conjecture is given.

[1]  Eitan Yaakobi,et al.  Error-Correction of Multidimensional Bursts , 2007, IEEE Transactions on Information Theory.

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

[3]  Gordon F. Royle,et al.  Algebraic Graph Theory , 2001, Graduate texts in mathematics.

[4]  Henry Cohn,et al.  The sphere packing problem in dimension 8The sphere packing problem in dimension 8 , 2016, 1603.04246.

[5]  Mihail N. Kolountzakis.  The Study of Translational Tiling with Fourier Analysis , 2004 .

[6]  Peter Horák,et al.  Tiling R5 by Crosses , 2014, Discret. Comput. Geom..

[7]  Bruce M. Kapron,et al.  The Cayley Graphs Associated With Some Quasi-Perfect Lee Codes Are Ramanujan Graphs , 2016, IEEE Transactions on Information Theory.

[8]  Reginaldo Palazzo Júnior,et al.  Quasi-Perfect Codes From Cayley Graphs Over Integer Rings , 2013, IEEE Transactions on Information Theory.

[9]  J. Lagarias,et al.  Structure of tilings of the line by a function , 1996 .

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

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

[12]  Timo Lepistö.  A Modification of the Elias-Bound and Nontexistence Theorems for Perfect Codes in the Lee-Metric , 1981, Inf. Control..

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

[14]  Paul M. Weichsel,et al.  Dominating sets in n-cubes , 1994, J. Graph Theory.

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

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

[17]  Jarkko Kari,et al.  An Algebraic Geometric Approach to Nivat's Conjecture , 2015, ICALP.

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

[20]  Sarit Buzaglo,et al.  Tilings by (0.5, n)-Crosses and Perfect Codes , 2013, SIAM J. Discret. Math..

[21]  Jeffrey C. Lagarias,et al.  Tiling the line with translates of one tile , 1996 .

[22]  Tao Zhang,et al.  Perfect and Quasi-Perfect Codes Under the $l_{p}$ Metric , 2017, IEEE Transactions on Information Theory.

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

[24]  Frank W. Barnes.  Algebraic theory of brick packing II , 1982, Discret. Math..

[25]  Henry Cohn,et al.  New upper bounds on sphere packings I , 2001, math/0110009.

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

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

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

[29]  Anxiao Jiang,et al.  Correcting Charge-Constrained Errors in the Rank-Modulation Scheme , 2010, IEEE Transactions on Information Theory.

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

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

[32]  Mario Szegedy,et al.  Algorithms to tile the infinite grid with finite clusters , 1998, Proceedings 39th Annual Symposium on Foundations of Computer Science (Cat. No.98CB36280).

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

[34]  Alexander Vardy,et al.  Interleaving Schemes for Multidimensional Cluster Errors , 1998, IEEE Trans. Inf. Theory.

[35]  Cristobal Camarero,et al.  Quasi-Perfect Lee Codes of Radius 2 and Arbitrarily Large Dimension , 2014, IEEE Transactions on Information Theory.

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

[37]  S. Bhattacharya.  Periodicity and decidability of tilings of $\mathbb{Z}^{2}$ , 2016, 1602.05738.

[38]  S. Bhattacharya.  Periodicity and decidability of tilings of ℤ2 , 2020, American Journal of Mathematics.

[39]  Peter Horák,et al.  A new approach towards the Golomb-Welch conjecture , 2014, Eur. J. Comb..

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

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

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

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

[45]  Jaakko Astola.  An Elias-type bound for Lee codes over large alphabets and its application to perfect codes , 1982, IEEE Trans. Inf. Theory.

[46]  Frank W. Barnes,et al.  Algebraic theory of brick packing I , 1982, Discret. Math..

[47]  Dongryul Kim.  Nonexistence of perfect 2-error-correcting Lee codes in certain dimensions , 2017, Eur. J. Comb..

[48]  Myung M. Bae,et al.  Resource Placement in Torus-Based Networks , 1997, IEEE Trans. Computers.

[49]  Sylvain Gravier,et al.  On the nonexistence of three-dimensional tiling in the Lee metric II , 2001, Discret. Math..

[51]  Factoring Elementary p-Groups for p ≤ 7 , 2011 .