Packing analogue of k-radius sequences

Let k be a positive integer. A sequence s1,s2,…,sm over an n-element A alphabet is a packing k-radius sequence, if for all pairs of indices (i,j), such that 1≤i0 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.