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Codes and Xor graph products Eyal Lubetzky

Summary: Codes and Xor graph products
Noga Alon
Eyal Lubetzky
November 22, 2005
What is the maximum possible number, f3(n), of vectors of length n over {0, 1, 2} such that
the Hamming distance between every two is even? What is the maximum possible number,
g3(n), of vectors in {0, 1, 2}n
such that the Hamming distance between every two is odd? We
investigate these questions, and more general ones, by studying Xor powers of graphs, focusing
on their independence number and clique number, and by introducing two new parameters of
a graph G. Both parameters denote limits of series of either clique numbers or independence
numbers of the Xor powers of G (normalized appropriately), and while both limits exist, one of
the series grows exponentially as the power tends to infinity, while the other grows linearly. As
a special case, it follows that f3(n) = (2n
) whereas g3(n) = (n).
1 Introduction
The Xor product of two graphs, G = (V, E) and H = (V , E ), is the graph whose vertex set
is the Cartesian product V V , where two vertices (u, u ) and (v, v ) are connected iff either
uv E, u v / E or uv / E, u v E , i.e., the vertices are adjacent in precisely one of their two


Source: Alon, Noga - School of Mathematical Sciences, Tel Aviv University
Lubetzky, Eyal - Theory Group, Microsoft Research


Collections: Computer Technologies and Information Sciences; Mathematics