Advanced search
Start date
(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

Full text
Qureshi, Claudio
Total Authors: 1
Document type: Journal article
Web of Science Citations: 1

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