Quasi-Perfect Lee Codes of Radius 2 and Arbitrarily Large Dimension

A construction of two-quasi-perfect Lee codes is given over the space ℤnp for p prime, p ≡ ±5 (mod 12), and n = 2[p/4]. It is known that there are infinitely many such primes. Golomb and Welch conjectured that perfect codes for the Lee metric do not exist for dimension n ≥ 3 and radius r ≥ 2. This conjecture was proved to be true for large radii as well as for low dimensions. The codes found are very close to be perfect, which exhibits the hardness of the conjecture. A series of computations show that related graphs are Ramanujan, which could provide further connections between coding and graph theories.

[1]  Tomás Vetrík,et al.  Abelian Cayley Graphs of Given Degree and Diameter 2 and 3 , 2014, Graphs Comb..

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

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

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

[5]  Ramón Beivide,et al.  Perfect Codes From Cayley Graphs Over Lipschitz Integers , 2009, IEEE Transactions on Information Theory.

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

[7]  Ioan Tabus,et al.  Bounds on the size of Lee-codes , 2013, 2013 8th International Symposium on Image and Signal Processing and Analysis (ISPA).

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

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

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

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

[12]  Giuliana P. Davidoff,et al.  Elementary number theory, group theory, and Ramanujan graphs , 2003 .

[13]  Ramón Beivide,et al.  Perfect Codes for Metrics Induced by Circulant Graphs , 2007, IEEE Transactions on Information Theory.

[14]  Stanislav Stankovic,et al.  On the use of Lee-codes for constructing multiple-valued error-correcting decision diagrams , 2012, 2012 5th International Symposium on Communications, Control and Signal Processing.

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

[16]  S. Nishimura,et al.  A generalization of the Lee distance and error correcting codes , 2008, Discret. Appl. Math..

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

[18]  Klaus Huber.  Codes over Gaussian integers , 1994, IEEE Trans. Inf. Theory.

[19]  Eitan Yaakobi,et al.  Coding for the Lee and Manhattan Metrics With Weighing Matrices , 2013, IEEE Transactions on Information Theory.

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

[21]  R. Goodstein,et al.  An introduction to the theory of numbers , 1961 .

[22]  Jozef Sirán,et al.  Cayley graphs of given degree and diameter for cyclic, Abelian, and metacyclic groups , 2012, Discret. Math..

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

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

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

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

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

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

[29]  J. Sirán,et al.  Moore Graphs and Beyond: A survey of the Degree/Diameter Problem , 2013 .

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

[31]  Ramón Beivide,et al.  Graph-based metrics over QAM constellations , 2008, 2008 IEEE International Symposium on Information Theory.

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