Abstract
Families of nonlattice tilings of ℝn by unit cubes are constructed. These tilings are specializations of certain families of nonlinear codes overGF(2). These cube-tilings provide building blocks for the construction of cube-tilings such that no two cubes have a high-dimensional face in common. We construct cube-tilings of ℝn such that no two cubes have a common face of dimension exceeding\(n - \tfrac{1}{3}\sqrt n\).
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K. Corrádi and S. Szabó, A combinatorial approach for Keller's conjecture,Period. Math. Hungar,21 (1990), 91–100.
G. Hajós, Über einfache und mehrfache Bedeckung desn-dimensionalen Raumes mit einem Würfelgitter,Math. Z. 47 (1942), 427–467.
G. Hajós, Sur la factorisation des groupes abelians,Časopis Pěst. Mat. Fys. 74 (1950), 157–162.
J. E. Hopcroft and J. D. Ullman,Introduction to Automata Theory, Languages and Computation, Addison-Wesley, Reading, MA, 1979.
O. H. Keller (1930), Über die lückenlose Einfüllung des Raumes mit Würfeln,J. Reine Angew. Math. 163 (1930), 231–248.
J. C. Lagarias and P. W. Shor, Keller's cube-tiling conjecture is false in high dimensions,Bull. Amer. Math. Soc., N.S. 27 (1992), 279–283.
H. R. Lewis and C. H. Papadimitriou,Elements of the Theory of Computation, Prentice-Hall, Englewood Cliffs, NJ, 1981.
J. MacWilliams and N. J. A. Sloane,The Theory of Error-Correcting Codes, North-Holland, Amsterdam, 1978.
H. Minkowski,Diophantische Approximationen, Teubner, Leipzig, 1907. Reprint: Physica-Verlag, Würzberg, 1961. (See Chapter 2, Section 4 and Chapter 3, Section 7. Minkowski's Conjecture appears on p. 28 and its geometric interpretation on p. 74).
O. Perron, Über lückenlose Ausfüllung desn-dimensionalen Raumes durch kongruente Würfel,Math. Z. 46 (1940), 1–26.
S. K. Stein, Algebraic Tiling,Amer. Math. Monthly 81 (1974), 445–462.
S. Szabó, A reduction of Keller's conjecture,Period. Math. Hungar. 17 (1986), 265–277.
J. H. van Lint,Introduction to Coding Theory, Springer-Verlag, New York, 1982.
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Lagarias, J.C., Shor, P.W. Cube-tilings of ℝn and nonlinear codesand nonlinear codes. Discrete Comput Geom 11, 359–391 (1994). https://doi.org/10.1007/BF02574014
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DOI: https://doi.org/10.1007/BF02574014