论文信息 - Expanders from symmetric codes

Expanders from symmetric codes

(MATH) A set S in the vector space FFpn is "good" if it satisfies the following (almost) equivalent conditions:<ul><li>S is an expanding generating set of Abelian group FFpn.</li><li>S are the rows of a generating matrix for a linear distance error-correcting code in FFpn.</li><li>All (nontrivial) Fourier coefficients of S are bounded by some ε ξ 1 (i.e. the set S is ε-biased). </li></ul>.A good set S must have at least cn vectors (with cρ1). We study conditions under which S is the orbit of only constant number of vectors, under the action of a finite group G on the coordinates. Such succinctly described sets yield very symmetric codes, and "amplifies" small constant-degree Cayley expanders to exponentially larger ones [19, 2].For the regular action (the coordinates are named by the elements of the group G), we develop representation theoretic conditions on the group G which guarantee the existence (in fact, abundance) of such few expanding orbits. The condition is a (nearly tight) upper bound on the distribution of dimensions of the irreducible representations of G, and is the main technical contribution of this paper. We further show a class of groups for which this condition is implied by the expansion properties of the group G itself! Combining these, we can iterate the amplification process above, and give (near-constant degree) Cayley expanders which are built from Abelian components.For other natural actions, such as of the affine group on a finite field, we give the first explicit construction of such few expanding orbits. In particular, we can completely derandomize the probabilistic construction of expanding generators in [2].

Avi Wigderson | Roy Meshulam

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