Constructions of asymptotically shortest k-radius sequences

Let k be a positive integer. A sequence s over an n-element alphabet A is called a k-radius sequence if every two symbols from A occur in s at distance of at most k. Let fk(n) denote the length of a shortest k-radius sequence over A. We provide constructions demonstrating that (1) for every fixed k and for every fixed ε>0, and (2) for every k=⌊nα⌋, where α is a fixed real such that 0<α<1, , for some β<2−α. Since , the constructions give asymptotically optimal k-radius sequences. Finally, (3) we construct optimal 2-radius sequences for a 2p-element alphabet, where p is a prime.