Abstract
Ac-ary Perfect Factor is a set of uniformly long cycles whose elements are drawn from a set of sizec, in which every possiblev-tuple of elements occurs exactly once. In the binary case, i.e. wherec=2, these perfect factors have previously been studied by Etzion [2], who showed that the obvious necessary conditions for their existence are in fact sufficient. This result has recently been extended by Paterson [4], who has shown that the necessary existence conditions are sufficient wheneverc is a prime power. In this paper we conjecture that the same is true for arbitrary values ofc, and exhibit a number of constructions. We also construct a family of related combinatorial objects, which we callPerfect Multi-factors.
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References
Burns, J., and Mitchell, C. J. 1993. Coding schemes for two-dimensional position sensing. In M. Ganley, editor,Cryptography and Coding III. Oxford University Press, pp. 31–66.
Etzion, T. 1988. Constructions for perfect maps and pseudo-random arrays.IEEE Transactions on Information Theory 34:1308–1316.
Mitchell, C. J., and Paterson, K. G. 1994. Decoding perfect maps.Designs, Codes and Cryptography 4:11–36.
Paterson, K. G. 1993. Perfect factors in the de Bruijn graph.Designs, Codes and Cryptography, to appear.
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Mitchell, C.J. Constructingc-ary Perfect Factors. Des Codes Crypt 4, 341–368 (1994). https://doi.org/10.1007/BF01388650
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DOI: https://doi.org/10.1007/BF01388650