22 research outputs found
Packing of R3 by crosses
The existence of tilings of R^n by crosses, a cluster of unit cubes comprising a central one and 2n arms, has been studied by several
authors. We have completely solved the problem for R^2, characterizing the crosses which lattice tile R^2, as well as determining the maximum packing density for the crosses which do not lattice tile the plane. In this paper we motivate a similar approach to study lattice packings of R^3 by crosses.The existence of tilings of Rn by crosses, a cluster of unit cubes comprising a central one and 2n arms, has been studied by several authors. We have completely solved the problem for R2 characterizing the crosses which lattice tile R2 as well as determining the maximum packing density for the crosses which do not lattice tile the plane. In this paper we motivate a similar approach to study lattice packings of R3 by crosses.publishe
Diameter Perfect Lee Codes
Lee codes have been intensively studied for more than 40 years. Interest in
these codes has been triggered by the Golomb-Welch conjecture on the existence
of the perfect error-correcting Lee codes. In this paper we deal with the
existence and enumeration of diameter perfect Lee codes. As main results we
determine all for which there exists a linear diameter-4 perfect Lee code
of word length over and prove that for each there are
uncountable many diameter-4 perfect Lee codes of word length over This
is in a strict contrast with perfect error-correcting Lee codes of word length
over \ as there is a unique such code for and its is
conjectured that this is always the case when is a prime. We produce
diameter perfect Lee codes by an algebraic construction that is based on a
group homomorphism. This will allow us to design an efficient algorithm for
their decoding. We hope that this construction will turn out to be useful far
beyond the scope of this paper