Abstract
A snake-in-the-box code (or snake) of word length n is a simple circuit in an n-dimensional cube Q n , with the additional property that any two non-neighboring words in the circuit differ in at least two positions. To construct such snakes a straightforward, non-recursive method is developed based on special linear codes with minimum distance 4. An extension of this method is used for the construction of covers of Q n consisting of 2m-1 vertex-disjoint snakes, for 2m-1 < n ≤ 2m. These covers turn out to have a symmetry group of order 2m.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Abbott H.L., Katchalski M. (1991) On the construction of snake-in-the-box codes. Utilitas Math. 40: 97–116
Alsardary S.Y. (1997) Further results on vertex covering of powers of complete graphs. Acta Math. Univ. Comenianae 66(2): 261–283
Casella D.A.: New lower bounds for the snake-in-the-box and the coil-in-the-box problems: using evolutionary techniques to hunt for snakes and coils. MSAI thesis (2005).
Casella D.A., Potter W.D.: New lower bounds for the snake-in-the-box problem: using evolutionary techniques to hunt for snakes. In: Proceedings of the 18th International Florida Artificial Intelligence Research Conference, pp. 264–269 (2005).
Emelyanov P.G., Lukito A. (2000) On the maximal length of a snake in a hypercube of small dimension. Discerete Math. 218: 51–59
Harary F.: Graph Theory, 3rd edn. Addison Wesley (1972).
Kautz W.H. (1958) Unit distance error checking codes. IEEE Trans. Electron. Comput. 7: 179–180
Klee V. (1967) A method for constructing circuit codes. J. Assoc. Comput. Mach. 14: 520–528
Knuth D.E.: The art of computer programming, vol. 4 Fascicle 2. Generating all tuples and permutations. Addison Wesley (2005).
Kochut K.J. (1996) Snake-in-the-box codes for dimension 7. J. Combin. Math. Combin. Comput. 20: 175–185
Lukito A.: Bounds for the length of certain types of distance preserving codes. Ph.D. thesis, Delft University of Technology (2000).
Lukito A., van Zanten A.J. (2001) A new non-asymptotic upper bound for snake-in-the-box codes. J. Combin. Math. Combin. Comput. 39: 147–156
MacWilliams F.J., Sloane N.J.A.: The Theory of Error Correcting Codes. North Holland Mathematical Library (1977).
Paterson K.G., Tuliani G. (1998) Some new circuit codes. IEEE Trans. Inform. Theory 44: 1305–1309
Potter W.D., et al.: Using the genetic algorithm to find snake-in-the-box codes. In: Proceedings of the 7th International Conference on Industrial and Engineering Applications of Artificial Intelligence and Expert Systems. United States Austin, Texas, pp. 421–426 (1994).
Snevily H.S. (1994) The snake-in-the-box problem, a new upper bound. Discrete Math. 133: 307–304
Solov’jeva F.I. (1987) An upper bound for the length of a cycle in an n-dimensional unit cube, Diskret. Analiz. 45: 71–76
Wojciechowski J. (1994) Covering the hypercube with a bounded number of disjoint snakes. Combinatorica 14: 491–496
van Zanten A.J. (1993) Minimal-change order and separability in linear codes. IEEE Trans. Inform. Theory 39: 1988–1989
van Zanten A.J., Lukito A. (1999) Construction of certain cyclic distance-preserving codes having linear-algebraic characteristic. Des. Codes Cryptogr. 16: 185–199
van Zanten A.J., Haryanto L.: Covers and Near-Covers of the Hypercube Q 16 by Symmetric Snakes, Report CS 06-01. Department of Computer Science, Universiteit Maastricht (2006).
Zémor G. (1997) An upper bound of the size of snakes. Combinatorica 17: 287–298
Open Access
This article is distributed under the terms of the Creative Commons Attribution Noncommercial License which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by V.A. Zinoviev.
Rights and permissions
Open Access This is an open access article distributed under the terms of the Creative Commons Attribution Noncommercial License (https://creativecommons.org/licenses/by-nc/2.0), which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.
About this article
Cite this article
van Zanten, A.J., Haryanto, L. Sets of disjoint snakes based on a Reed-Muller code and covering the hypercube. Des. Codes Cryptogr. 48, 207–229 (2008). https://doi.org/10.1007/s10623-008-9202-x
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10623-008-9202-x