On Canonical Gray Cycles

Stanislav Sykora,

Extra Byte, Via R.Sanzio 22C, 20022 Castano Primo (MI), Italy,
in Stan's Library, Vol.V. First release 31 January 2014

Permalink:  DOI: 10.3247/SL5Math14.001

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This article explores cycles of integers sharing the special property of the ?classical? Gray sequence which make the latter so popular in technical applications, namely the fact that the binary expansions of any two adjacent members differ by exactly one digit. However, there are many cycles having this property: given any even number 2n > 0, it is always possible to find at least one permutation of numbers 0 to 2n which has it. Even accounting for symmetry-induced equivalences (such as rotating or reversing the cycle) , there are for many values of n multiple, genuinely distinct Gray cycles of length 2n, some of which with a potential for useful practical applications. This article focuses on counting the equivalence classes of binary Gray cycles, identified with a set of canonical Gray cycles (CGC). Using a brute-force enumeration algorithm, the number of CGC's was computed up to the cycle length of 32.

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Please, cite this online document as:
Sykora S., On Canonical Gray Cycles,
Stan's Library, Vol.V, January 2014, DOI: 10.3247/SL5Math14.001 .


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Copyright ©2014 Stan's Library, ISSN 2421-1230, DOI: 10.3247/SL5Math14.001 Designed by Stan Sýkora