Packing analogue of k-radius sequences

Let k be a positive integer. A sequence s₁, s₂,.., s_m over an n-element A alphabet is a packing k-radius sequence, if for all pairs of indices (i, j), such that 1≤i<j≤m and j-i≤k, the sets {s_i, s_j} are pairwise different 2-element subsets of A. Let g_k(n) denote the length of a longest k-radius sequence over A. We give a construction demonstrating that for every k=⌊cn^α⌋, where c and α are fixed reals such that c>0 and 0≤α<1/2, g_k(n)=n²/2k(1-o(1)). For a constant k we show that g_k(n)=n²/2k-O(n^1.525). Moreover, we prove an upper bound for g_k(n) that allows us to show that g_k(n)=n(1+o(1)) for every k=⌊cn^α⌋, where c>0 and 1/2<α<1.