This Article 
   
 Share 
   
 Bibliographic References 
   
 Add to: 
 
Digg
Furl
Spurl
Blink
Simpy
Google
Del.icio.us
Y!MyWeb
 
 Search 
   
Mixed-Radix Gray Codes in Lee Metric
October 2007 (vol. 56 no. 10)
pp. 1297-1307
Gray codes, where two consecutive codewords differ in exactly one position by $\pm 1$,are given. In a single radix code, all dimensions have the same base, say $k$, whereas in a mixed radix code the base in one dimension can be different from the base in another dimension. Constructions of new classes of mixed radix Gray codes are presented. It is shown how these codes can be used as a basis for constructing edge disjoint Hamiltonian cycles in mixed radix toroidal networks when the number of dimensions, $n=2^r$ for some $r \geq 0$. Efficient algorithms for the generation of these codes are then shown.

[1] D.J. Amalraj, N. Sundararajan, and G. Dhar, “A Data Structure Based on Gray Code Encoding for Graphics and Image Processing,” Proc. SPIE: Applications of Digital Image Processing XIII, pp.65-76, 1990.
[2] M. Bae and B. Bose, “Lee Distance Gray Codes and Edge Disjoint Hamiltonian Cycles in Toroidal Networks,” Proc. 14th IEEE Int'l Parallel and Distributed Processing Symp. (IPDPS '00), pp. 365-370, May 2000.
[3] J.R. Bitner, G. Ehrlich, and E.M. Reingold, “Efficient Generation of the Binary Reflected Gray Code and Its Applications,” Comm. ACM, vol. 19, no. 9, Sept. 1976.
[4] B. Bose, R. Broeg, Y. Kwon, and Y. Ashir, “Lee Distance and Topological Properties of $k$ -Ary $n$ -Cubes,” IEEE Trans. Computers, vol. 44, no. 8, pp. 1021-1030, Aug. 1995.
[5] R. Broeg, B. Bose, and V. Lo, “Lee Distance, Gray Codes, and the Torus,” Telecomm. Systems, vol. 10, pp. 21-32, 1998.
[6] M. Cohn, “Affine $m$ -Ary Gray Codes,” Information and Control, vol. 6, pp. 70-78, 1963.
[7] C.C. Chang, H.Y. Chen, and C.Y. Chen, “Symbolic Gray Codes as a Data Allocation Scheme for Two Disc Systems,” Computer J., vol. 35, no. 3, pp. 299-305, 1992.
[8] M. Chen and K.G. Shin, “Subcube Allocation and Task Migration in Hypercube Machines,” IEEE Trans. Computers, vol. 39, no. 9, pp.1146-1155, Sept. 1990.
[9] P. Diaconis and S. Holmes, “Gray Codes for Randomization Procedures,” Statistics and Computing, vol. 4, pp. 287-302, 1994.
[10] D.E. Knuth, The Art of Computer Programming: Fascicle 2: Generating All Tuples and Permutations, vol. 4. Addison-Wesley, 2005.
[11] M.C. Er, “On Generating $N$ -Ary Reflected Gray Codes,” IEEE Trans. Computers, vol. 33, no. 8, pp. 739-741, Aug. 1984.
[12] C. Faloutsos, “Gray Codes for Partial Match and Range Queries,” IEEE Trans. Software Eng., vol. 14, no. 10, pp. 1381-1393, Oct. 1988.
[13] M. Gardner, “Curious Properties of the Gray Code and How It Can Be Used to Solve Puzzles,” Scientific Am., vol. 227, no. 2, pp.106-109, 1972.
[14] E.N. Gilbert, “Gray Codes and Paths on the $n$ -Cube,” Bell Systems Technical J., vol. 37, pp. 815-826, 1958.
[15] F. Gray, “Pulse Cose Communication,” US patent 2632058, Mar. 1953.
[16] R.M. Losee, “A Gray Code Based Ordering for Documents on Shelves: Classification for Browsing and Retrieval,” J. Am. Soc. Information Science, vol. 43, no. 4, pp. 312-322, 1992.
[17] J.E. Ludman, “Gray Code Generation for MPSK Signals,” IEEE Trans. Comm., vol. 29, pp. 1519-1522, 1981.
[18] A. Nijenhuis and H.S. Wilf, Combinatorial Algorithms for Computers and Calculators. Academic Press, 1978.
[19] E. Reingold, J. Nievergelt, and N. Deo, Combinatorial Algorithms —Theory and Practise. Prentice Hall, 1977.
[20] D. Richards, “Data Compression and Graycode Sorting,” Information Processing Letters, vol. 22, pp. 205-210, 1986.
[21] J. Robinson and M. Cohn, “Counting Sequences,” IEEE Trans. Computers, vol. 30, no. 1, pp. 17-23, Jan. 1981.
[22] C. Savage, “A Survey of Combinatorial Gray Codes,” SIAM Rev., vol. 39, no. 4, pp. 605-629, Dec. 1997.
[23] H.S. Wilf, Combinatorial Algorithms: An Update. SIAM, 1989.

Index Terms:
Lee Distance, Gray Code, Hamiltonian Cycle, Toroidal Networks
Citation:
Madhusudhanan Anantha, Bella Bose, Bader AlBdaiwi, "Mixed-Radix Gray Codes in Lee Metric," IEEE Transactions on Computers, vol. 56, no. 10, pp. 1297-1307, Oct. 2007, doi:10.1109/TC.2007.1083
Usage of this product signifies your acceptance of the Terms of Use.