Expanders from symmetric codes

A set S in the vector space F_pⁿ is "good" if it satisfies any of the following (almost) equivalent conditions: (1) S are the rows of a generating matrix for a linear distance code, (2) all (nontrivial) Fourier coefficients of S are bounded away from 1, and (3) the Cayley graph on F_p ⁿ with generators S is a good expander A good set S must have at least cn vectors (with c > 1). We study conditions under which S is the orbit of only a constant number of vectors, under the action of a finite group G on the n coordinates. Such succinctly described sets yield very symmetric codes, and can "amplify" small constant-degree Cayley expanders to exponentially larger ones. For the regular action (the coordinates are named by the elements of the group G), we develop representative 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! By 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