Summary: Discrete Applied
Mathematics 37138 (1992) 9-11
Transmitting in the n-dimensional
Department of Mathematics, Sackler Faculty of Exact Sciences, Tel Aviv University, Tel Aviv, Israel; and
BeNCore, 445 South Street, Morristown, NJ 07960, USA.
Received 1 July 1989
Alon, N., Transmitting in the n-dimensional cube, Discrete Applied Mathematics 37/38 (1992) 9-1 I.
Motivated by a certain communication problem we show that for any integer n and for any sequence
(a,,...,ak) of k = [n/21 binary vectors of length n, there is a binary vector z of length n whose Hamming
distance from a, is strictly bigger than k-i for all 15 i 5 k.
The n-dimensional cube is the graph whose vertices are all 2" binary vectors of
length n, in which two vertices are adjacent if and only if their Hamming distance
is 1, i.e., they differ in precisely one coordinate. Suppose there is a processor on
each of these vertices and suppose there is an additional entity, outside the cube,
which we call here the Sender, that has a message which has to be transmitted to
all the processors on the cube. The transmission should be carried out as follows;