L_1-norm packings from function fields

In this paper,we study some packings in a cube,namely,how to pack n points in a cube so as to maximize the minimal distance.The distance is induced by the L_1-norm which is analogous to the Hamming distance in coding theory.Two constructions with reasonable parameters are obtained,by using some results from a function field including divisor class group,narrow ray class group,and so on.We also present some asymptotic results of the two packings.     

The Möbius invariants for circle packings

Möbius invariants of general circle packings are defined in terms of cross ratios. The necessary and sufficient conditions of existence of circle packings are established by the techniques of Möbius invariants. It is shown that circle packings are uniquely determined, up to Möbius transformations, by their Möbius invariants. The rigidity of infinite circle packings with bounded degree is proved using the approach of Möbius invariants.     

Directed packings with block size 5 and even ν

Let andk be positive integers. A transitively orderedk-tuple (a ₁,a ₂,...,a _k) is defined to be the set {(a _i, a_j) 1i<jk} consisting ofk(k–1)/2 ordered pairs. A directed packing with parameters ,k and index =1, denoted byDP(k, 1; ), is a pair (X, A) whereX is a -set (of points) andA is a collection of transitively orderedk-tuples ofX (called blocks) such that every ordered pair of distinct points ofX occurs in at most one block ofA. The greatest number of blocks required in aDP(k, 1; ) is called packing number and denoted byDD(k, 1; ). It is shown in this paper that for all even integers , where [x] is the floor ofx.     

Schwarz’s lemma for the circle packings with the general combinatorics

Rodin (1987) proved the Schwarz’s lemma analog for the circle packings based on the hexagonal combinatorics. In this paper, we prove the Schwarz’s lemma for the circle packings with the general combinatorics and our proof is more simpler than Rodin’s proof. At the same time, we obtain a rigidity property for those packings with the general combinatorics.     

Minimal coverings and maximal packings of (k−1)-subsets by k-subsets

This paper studies the asymptotic behavior of functions M(n, k, k–1, ) and m(n, k, k–1, ), equal to the respective cardinalities of the minimal -covering and maximal -packing of all (k–1)-subsets of the n-element set of its k-subsets. It is shown that, if sequence k=k(n) is such that k(n)/n 0 as then m(n, k, k–1, ).( _k–1 ⁿ ).k¹, and if as n , thenM(n,k,k–1,).( _k–1 ⁿ ).k^–1. A consequence of these results is the validity of the Erdös-Hanani conjecture concerning the asymptotic behavior of functions M(n, k, k–1, 1) and m(n, k, k–1, 1).Translated from Matematicheskie Zametki, Vol. 21, No. 4, pp. 565–571, April, 1977.In conclusion, the author wishes to thank A. A. Sapozhenko under whose direction the present work was achieved.     

Dense lattice packings of spheres in Euclidean spaces of dimension n⩽16 *

We present a number of lattice packings of equal spheres in ⁿ for n16 For n15, these packings have the same density as the densest known lattice packings. For n=16, the packing described here is denser than the known ones.It should be pointed out that the 16-dimensional lattice described here is equivalent to one found by E. S. Barnes and G. E. Wall, J. Aust. Math. Soc.,1, 47–63 (1959); see also J. Leech and N. J. A. Sloane, Can. J. Math.,23, 718–745 (1971) — Translator.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 82, 144–146, 1979.