Packing analogue of [formula omitted]-radius sequences

Abstract

Let k be a positive integer. A sequence s1,s2,…,sm over an n-element A alphabet is a packingk-radius sequence, if for all pairs of indices (i,j), such that 1≤i<j≤m and j−i≤k, the sets {si,sj} are pairwise different 2-element subsets of A. Let gk(n) denote the length of a longest k-radius sequence over A. We give a construction demonstrating that for every k=⌊cnα⌋, where c and α are fixed reals such that c>0 and 0≤α<12, gk(n)=n22k(1−o(1)). For a constant k we show that gk(n)=n22k−O(n1.525). Moreover, we prove an upper bound for gk(n) that allows us to show that gk(n)=n(1+o(1)) for every k=⌊cnα⌋, where c>0 and 12<α<1.