Constructions of asymptotically shortest k-radius sequences

TL;DR

This paper presents new constructions for k-radius sequences that are asymptotically shortest, providing bounds and explicit optimal sequences for specific cases, advancing understanding of sequence minimality in combinatorics.

Contribution

The paper introduces constructions demonstrating asymptotic optimality of k-radius sequences for fixed and varying k, including explicit optimal sequences for certain prime-based alphabets.

Findings

f_k(n) = n^2/(2k) + O(n^(1+e)) for fixed k and e>0

f_k(n) = n^2/(2k) + O(n^b) for k ~ n^a, 0<a<1

Constructed optimal 2-radius sequences for 2p-element alphabets where p is prime

Abstract

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 f_k(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 e>0, f_k(n) = n^2/(2k) +O(n^(1+e)) and (2) for every k, where k is the integer part of n^a for some fixed real a such that 0 < a <1, f_k(n) = n^2/(2k) +O(n^b), for some b <2-a. Since f_k(n) >= n^2/(2k) - n/(2k), 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.