Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Diameter Perfect Lee Codes   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 $q$ for which there exists a linear diameter-4 perfect Lee code
of word length $n$ over $Z_{q},$ and prove that for each $n\geq 3$ there are
uncountable many diameter-4 perfect Lee codes of word length $n$ over $Z.$ This
is in a strict contrast with perfect error-correcting Lee codes of word length
$n$ over $Z\,$\ as there is a unique such code for $n=3,$ and its is
conjectured that this is always the case when $2n+1$ 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.

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The categories of this paper are: mtrl-th
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The categories of this paper are: bayes-an
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The categories of this paper are: solv-int
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The categories of this paper are: cond-mat
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The categories of this paper are: atom-ph
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The categories of this paper are: q-fin
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The categories of this paper are: acc-phys
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The categories of this paper are: gr-qc
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The categories of this paper are: math-ph
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The categories of this paper are: ao-sci
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The categories of this paper are: nucl-ex
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The categories of this paper are: hep-ph
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The categories of this paper are: supr-con
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The categories of this paper are: stat
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The categories of this paper are: econ
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The categories of this paper are: hep-lat