Sequences of large radius

A k -radius sequence over an n -element alphabet A is a sequence in which every two elements of A occur within distance k of each other, where the distance is defined as the difference of indices of terms. The problem of finding (for given k and n ) such sequences of a small length, motivated by some problems occurring in large data transfer, has been investigated by several authors recently. In this paper we consider a slight modification of the problem. We change the definition of the distance between terms of the sequence by computing it as if the terms were placed on a circle. The length of the shortest such sequence is denoted by g k ( n ) . estigate sequences of radius linear in the alphabet size. For some values of c < 1 and sufficiently large n we find the exact values of g c n ( n ) by providing direct constructions of optimal sequences. In the constructions we use Singer difference sets. Moreover, we give a nontrivial general lower bound for g c n ( n ) .