On distance Gray codes

On distance Gray codes

A Gray code of size n is a cyclic sequence of all binary words of length n such that two consecutive words differ exactly in one position. We say that the Gray code is a distance code if the Hamming distance between words located at distance k from each other is equal to d . The distance property ge...

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Bibliographic Details
Published in:Journal of applied and industrial mathematics Vol. 11; no. 2; pp. 185 - 192
Main Authors: Bykov, I. S., Perezhogin, A. L.
Format: Journal Article
Language:English
Published: Moscow Pleiades Publishing 01-04-2017
Springer Nature B.V
Subjects:
Binary codes
Codes
Infinite series
Mathematical analysis
Mathematics
Mathematics and Statistics
uniform Gray code
Gray code
cube
Hamiltonian cycle
antipodal Gray code
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Summary:A Gray code of size n is a cyclic sequence of all binary words of length n such that two consecutive words differ exactly in one position. We say that the Gray code is a distance code if the Hamming distance between words located at distance k from each other is equal to d . The distance property generalizes the familiar concepts of a locally balanced Gray code. We prove that there are no distance Gray codes with d = 1 for k > 1. Some examples of constructing distance Gray codes are given. For one infinite series of parameters, it is proved that there are no distance Gray codes.
ISSN:1990-4789
1990-4797
DOI:10.1134/S1990478917020041