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(Reference retrieved automatically from Web of Science through information on FAPESP grant and its corresponding number as mentioned in the publication by the authors.)

On the non-existence of linear perfect Lee codes: The Zhang-Ge condition and a new polynomial criterion

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Author(s):
Qureshi, Claudio
Total Authors: 1
Document type: Journal article
Source: EUROPEAN JOURNAL OF COMBINATORICS; v. 83, JAN 2020.
Web of Science Citations: 1
Abstract

The Golomb-Welch conjecture (1968) states that there are no e-perfect Lee codes in Z(n) for n >= 3 and e >= 2. This conjecture remains open even for linear codes. A recent result of Zhang and Ge establishes the non-existence of linear e-perfect Lee codes in Z(n) for infinitely many dimensions n, for e = 3 and 4. In this paper we extend this result in two ways. First, using the non-existence criterion of Zhang and Ge together with a generalized version of Lucas' theorem we extend the above result for almost all e (i.e. a subset of positive integers with density 1). Namely, if e contains a digit 1 in its base-3 representation which is not in the unit place (e.g. e = 3, 4) there are no linear e-perfect Lee codes in Z(n) for infinitely many dimensions n. Next, based on a family of polynomials (the Q-polynomials), we present a new criterion for the non-existence of certain lattice tilings. This criterion depends on a prime p and a tile B. For p = 3 and B being a Lee ball we recover the criterion of Zhang and Ge. (C) 2019 Elsevier Ltd. All rights reserved. (AU)

FAPESP's process: 15/26420-1 - Metrics in the context of information theory and error correcting codes
Grantee:Claudio Michael Qureshi Valdez
Support type: Scholarships in Brazil - Post-Doctorate
FAPESP's process: 13/25977-7 - Security and reliability of Information: theory and practice
Grantee:Marcelo Firer
Support type: Research Projects - Thematic Grants