Perfect codes in the lp metric

We investigate perfect codes in Z n in the ? p metric. Upper bounds for the packing radius r of a linear perfect code, in terms of the metric parameter p and the dimension n are derived. For p = 2 and n = 2 , 3 , we determine all radii for which there exist linear perfect codes. The non-existence results for codes in Z n presented here imply non-existence results for codes over finite alphabets Z q , when the alphabet size is large enough, and have implications on some recent constructions of spherical codes.

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