| Full text | |
| Author(s): |
Qureshi, Claudio
Total Authors: 1
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| 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 Opportunities: | Scholarships in Brazil - Post-Doctoral |
| FAPESP's process: | 13/25977-7 - Security and reliability of Information: theory and practice |
| Grantee: | Marcelo Firer |
| Support Opportunities: | Research Projects - Thematic Grants |