Paper Info
Title
On the non-existence of linear perfect Lee codes: The Zhang-Ge condition and a new polynomial criterion.
Abstract
The Golomb–Welch conjecture (1968) states that there are no e-perfect Lee codes in Zn 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 Zn 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 Zn 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.
Year
DOI
Venue
2018
10.1016/j.ejc.2019.103022
European Journal of Combinatorics
Field
DocType
Volume
Integer,Prime (order theory),Discrete mathematics,Polynomial,Lattice (order),Conjecture,Zhàng,Mathematics
Journal
83
ISSN
Citations 
PageRank 
0195-6698
0
0.34
References 
Authors
6
1
Name
Order
Citations
PageRank
Claudio Qureshi1104.48