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In this lecture, I will explain how to make graphs from linear error-correcting codes. These will come close to being expanders, except that their degrees are logarithmic rather than constant. In preparation for constructing expanders in the next lecture, I will show how one can improve the expansion of a graph by squaring it. If there is time, I will say a little about Cayley graphs in general. 15.2 Graphs from Linear Codes Consider a linear code over {0, 1} from m bits to n bits. We may assume that such a code is encoded by an n-by-m matrix G, and that its codewords are the vectors Gb, where b ∈ {0, 1} m. Let d be the minimum distance of this code (warning: in this section d is not degree). We will use this code to construct an n-regular graph on 2 m vertices with λ2 = 2d. The construction will be a generalization of the hypercube, and we in fact obtain the hypercube we set G = I m. We will take as the vertex set V = {0, 1} m. Thus, I will also write vertices as vectors, such as x and y. Two vertices x and y will be connected by an edge if their sum modulo 2 is a row of G. Let me say that again. Let g 1,..., g n be the rows of G. Then, the graph has edge set (x, x + g j) : x ∈ V, 1 ≤ j ≤ n}. Of course, this addition is taken modulo 2. You should now verify that if G is the identity matrix, we get the hypercube. In the general case, it is like a hypercube with extra edges. This graph is a Cayley graph over the additive group (Z/2Z) m: that is the set of strings in {0, 1} m under addition modulo 2. Other Cayley graphs that we have seen in this class include the hypercubes, the ring graphs, and the Payley graphs. In fact, these are all Cayley graphs over Abelian groups. The great thing about Cayley graphs over Abeliean groups is that their eigenvectors are determined just from the group 1. They do not depend upon the choice of generators. Knowing 1 More precisely, the characters always form an orthonromal set of eigenvectors, and the characters just depend upon the group. When two different characters have the same eigenvalue, we obtain an eigenspace of dimension greater than 1. These eigenspaces do depend upon the choice of generators

Year: 2012

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