English version:
Journal of Applied and Industrial Mathematics, 2017, 11:2, 185-192

Volume 24, No 2, 2017, P. 5-17

UDC 519.17
I. S. Bykov and A. L. Perezhogin
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 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.
Tab. 5, bibliogr. 9.

Keywords: $n$-cube, Hamiltonian cycle, Gray code, uniform Gray code, antipodal Gray code.

DOI: 10.17377/daio.2017.24.545

Igor S. Bykov 2
Alexey L. Perezhogin 1,2

1. Sobolev Institute of Mathematics,
4 Koptyug Ave., 630090 Novosibirsk, Russia
2. Novosibirsk State University,
2 Pirogov St., 630090 Novosibirsk, Russia
e-mail: patrick.no10@gmail.com, pereal@math.nsc.ru

Received 19 May 2016
Revised 16 September 2016


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 © Sobolev Institute of Mathematics, 2015