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

02/23/2023

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by   Viviana Gubitosi, et al.

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In this paper we consider codes in 𝔽_q^s× r with packing radius R regarding the NRT-metric (i.e. when the underlying poset is a disjoint union of chains with the same length) 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 non-trivial 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.