On the divisibility of the rank of appearance of a Lucas sequence

Let U = (U-n)(n >= 0) be a Lucas sequence and, for every prime number p, let rho(U)(p) be the rank of appearance of p in U, that is, the smallest positive integer k such that p divides U-k, whenever it exists. Furthermore, let d be an odd positive integer. Under some mild hypotheses, we prove an asymptotic formula for the number of primes p <= x such that d divides p(U)(p), as x -> +infinity

On the Existence of Optimal Exact-Repair MDS Codes for Distributed Storage

The high repair cost of (n,k) Maximum Distance Separable (MDS) erasure codes has recently motivated a new class of codes, called Regenerating Codes, that optimally trade off storage cost for repair bandwidth. In this paper, we address bandwidth-optimal (n,k,d) Exact-Repair MDS codes, which allow for any failed node to be repaired exactly with access to arbitrary d survivor nodes, where k<=d<=n-1. We show the existence of Exact-Repair MDS codes that achieve minimum repair bandwidth (matching the cutset lower bound) for arbitrary admissible (n,k,d), i.e., k<n and k<=d<=n-1. Our approach is based on interference alignment techniques and uses vector linear codes which allow to split symbols into arbitrarily small subsymbols.Comment: 20 pages, 6 figure