On the non-existence of perfect codes in the NRT-metric

In this paper we consider codes in F s×r q with packing radius R regarding the NRT-metric (i.e. when the underlying poset is a disjoint union of s chains with the same length r) and we establish necessary condition on the parameters s, r and R for the existence of perfect codes. More explicitly, for r, s ≥ 2 and R ≥ 1 we prove that if there is a nontrivial perfect code then (r+1)(R+1) ≤ rs. We also explore a connection to the knapsack problem and establish a correspondence between perfect codes with r > R and those with r = R. Using this correspondence we prove the non-existence of non-trivial perfect codes also for s = R + 2.