17 research outputs found

    Diameter Perfect Lee Codes

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    Lee codes have been intensively studied for more than 40 years. Interest in these codes has been triggered by the Golomb-Welch conjecture on the existence of the perfect error-correcting Lee codes. In this paper we deal with the existence and enumeration of diameter perfect Lee codes. As main results we determine all qq for which there exists a linear diameter-4 perfect Lee code of word length nn over Zq,Z_{q}, and prove that for each n3n\geq 3 there are uncountable many diameter-4 perfect Lee codes of word length nn over Z.Z. This is in a strict contrast with perfect error-correcting Lee codes of word length nn over ZZ\,\ as there is a unique such code for n=3,n=3, and its is conjectured that this is always the case when 2n+12n+1 is a prime. We produce diameter perfect Lee codes by an algebraic construction that is based on a group homomorphism. This will allow us to design an efficient algorithm for their decoding. We hope that this construction will turn out to be useful far beyond the scope of this paper

    On perfect 1-mathcalEmathcal E-error-correcting codes

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    We generalize the concept of perfect Lee-error-correcting codes, and present constructions of this new class of perfect codes that are called perfect 1-mathcalEmathcal{E}-error-correcting codes. We also show that in some cases such codes contain quite a few perfect 1-error-correcting qq-ary Hamming codes as subsets
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