Two Results on the Bit Extraction Problem (extended Abstract)

We study the problem of coloring the n-dimensional boolean cube with c = 2 s colors such that in every k-dimensional subcube each color appears 2 k =c times. We are interested in the smallest k for which such a coloring exists. This problem is equivalent to the t-resilient function problem. A coloring is locally symmetric if each vertex in the n-cube has the same number of neighbors from each color other than its own. An XOR-type coloring is one obtained by a linear map GF (2) n ! GF (2) s. Our main results answer the following problems proposed by Friedman 4]: 1) if c ? 1jn, are all optimal colorings necessarily locally symmetric? 2) are there optimal colorings not obtainable as XOR type colorings? We give positive answers to both questions. Along the way we prove some identities involving the distance distribution of color classes, in order to gain a better understanding of the structure of color classes. Our main tool is an expression of the eigenvalues of the distance-i adjacency matrix of the n-cube via the Krawtchouk polynomials.