Short k‐radius sequences, k‐difference sequences and universal cycles

An n‐ary k‐radius sequence is a finite sequence of elements taken from an alphabet of size n in which any two distinct elements occur within distance k of each other somewhere in the sequence. The study of constructing short k‐radius sequences was motivated by some problems occurring in large data transfer. Let f k ( n ) be the shortest length of any n‐ary k‐radius sequence. We show that the conjecture f k ( n ) = n 2 2 k + O ( n ) by Bondy et al is true for k ≤ 4, and determine the exact values of f 2 ( n ) for new infinitely many n. Further, we investigate new sequences which we call k‐difference, as they are related to k‐radius sequences and seem to be interesting in themselves. Finally, we answer a question about the optimal length of packing and covering analogs of universal cycles proposed by Dębski et al.