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