- Email: [email protected]
Packings with large minimum kissing numbers
DISCRETE MATHEMATICS ELSEVIER
Discrete Mathematics 175 (1997) 249 251
Note
Packings with large minimum kissing numbers Noga Alon l Department of Mathematics, Raymond and Beverly Sackler Faculo~ o! Exact Sciences. Tel Aviv University, Tel At,iv, Ls'rael Received 10 December 1996; revised 24 February 1997
Abstract
For each proper power of 4, n, we describe a simple explicit construction of a finite collection of pairwise disjoint open unit balls in R~ in which each ball touches more than 2 '/'~ others. @ 1997 Elsevier Science B.V.
A p a c k & 9 o f balls in the Euclidean space is a finite or infinite collection o f pairwise disjoint open unit balls in R ' . It is called a lattice packing if the centers o f the balls form a lattice in R". The minimum kissing number of a packing is the minimum number o f balls touching a given one. Note that for a lattice packing this is simply the number o f balls touching any given one, since every ball touches the same number of others. The problem o f existence and construction o f lattice packings with high kissing numbers received a considerable amount o f attention, and there are several known constructions that show that the kissing number of a lattice packing o f balls in ~" may be at least n ~(l°gn) = 2 (2(l°g2n). See [ 2 - 6 ] . The problem o f constructing finite packings with a large minimum kissing number received much less attention. In this short note we consider this problem and construct finite packings in [R~ with much higher kissing numbers than those o f the known lattice packings. For each (proper) power of 4, n the kissing number o f our packing in R" exceeds 2 '/~. To do so, we construct, for each integer k>~2, a linear, binary, error correcting code o f length n = 4 k, dimension k(2 k l + 1) and minimum distance n/4 in which the number of words of minimum Hamming weight is
(2k)
(2 a - 1) 2~_1
I
> 2 2k = 2 ~/n.
E-mail: [email protected].
0012-365X/97/$17.00 Copyright @ 1997 Elsevier Science B.V. All rights reserved PII SO0 I 2-365X(97)0007 I-X
N. Alon/DiscreteMathematics175 (1997) 249-251
250
It is easy to check that the finite collection of balls o f radius x/~/4 centered at all the vectors o f this linear code (considered as real vectors) forms a finite packing o f balls whose minimum kissing number is (2 k - l )
2k_i
> 2 2k = 2 v~.
Our construction is a concatenation code which is a variant o f one o f the constructions in [1]. Here are the details. Let F = GF(2 k) be the finite field with 2 k elements, where each member of F is represented by a binary vector o f length k. Construct the generating matrix A o f a linear code over GF(2) as follows. A has k(2 k-1 + 1) rows and 4 k columns. Let vl . . . . . v~ be a basis o f GF(2 ~) over GF(2). The rows o f A are indexed by pairs (i,j) with 0~
k
i=0 j=l
Note that the sum 2k-I
k
Z Z Ei,jl)jXi i=0 j= 1
is simply a polynomial of degree at most 2 k-1 over F, and therefore it is zero for at most 2 k-1 different values of x in F. If it is not zero for some x then as y varies exactly half o f the elements y give a 0 inner product with it and exactly half give a 1 product. Thus the Hamming weight is determined by the number o f zeros (in F ) o f the polynomial. There are only 2 k-I + 1 distinct weights and the minimum weight is obtained from polynomials with 2 k-l roots. The number o f such polynomials is precisely (2 k - l )
2k_t
'
where the 2 k - 1 factor comes from the fact that the polynomials are not necessarily monic, and the binomial coefficient is the number o f ways to choose the roots o f the polynomial. This proves the desired result and hence supplies the desired packings as well. Note that by letting y in the construction above vary over F - {0} instead of F we get a related code o f length n = 4 k - 2 k and the same dimension and weight distribution as the above one. Note also that by changing the upper bound for i in the definition o f the generating matrix A from 2 k-I to an arbitrary parameter h ( < 2 k) we get other codes with interesting parameters. A similar construction works, of course, for fields of other characteristics.
N. Alon/Discrete Mathematics 175 ( 1 9 9 7 ) 2 4 9 ~ 5 1
251
It would be interesting to decide if there is a linear binary code of length n in which the number of words of minimum Hamming weight is 2 al"). It would also be interesting to decide if there is a lattice packing in R" whose kissing number is 2 Q°'). We strongly suspect that such codes and such lattice packings do exist.
Acknowledgements I would like to thank N. Sloane for pointing out an error in a previous version of this note and for helpful comments.
References [1] N. Alon, O. Goldreich, J. Hastad, R. Peralta, Simple constructions of almost k-wise independent random variables, in: Proc. 31st IEEE FOCS, St. Louis, Missouri, IEEE, 1990, pp. 544-553; also in: Random Structures and Algorithms 3 (1992) 289-304. [2] E.S. Barnes, N.J.A. Sloane, New lattice packings of spheres, Canad. J. Math. 35 (1983) 30-41. [3] E.S. Barnes, G.E. Wall, Some extreme forms defined in terms of Abelian groups, J. Austral. Math. Soc. I (1959) 47~53. [4] A. Bos, J.H. Conway, N.J.A. Sloane, Further lattice packings in high dimensions, Mathematika 29 (1982) 171-180. [5] J.H. Conway, N.J.A. Sloane, Sphere Packings, Lattices and Groups, Springer, Berlin, 1988. [6] J. Leech, Some sphere packings in higher space, Canad. J. Math. 16 (1984) 657~582.
Discrete Mathematics 175 (1997) 249 251
Note
Packings with large minimum kissing numbers Noga Alon l Department of Mathematics, Raymond and Beverly Sackler Faculo~ o! Exact Sciences. Tel Aviv University, Tel At,iv, Ls'rael Received 10 December 1996; revised 24 February 1997
Abstract
For each proper power of 4, n, we describe a simple explicit construction of a finite collection of pairwise disjoint open unit balls in R~ in which each ball touches more than 2 '/'~ others. @ 1997 Elsevier Science B.V.
A p a c k & 9 o f balls in the Euclidean space is a finite or infinite collection o f pairwise disjoint open unit balls in R ' . It is called a lattice packing if the centers o f the balls form a lattice in R". The minimum kissing number of a packing is the minimum number o f balls touching a given one. Note that for a lattice packing this is simply the number o f balls touching any given one, since every ball touches the same number of others. The problem o f existence and construction o f lattice packings with high kissing numbers received a considerable amount o f attention, and there are several known constructions that show that the kissing number of a lattice packing o f balls in ~" may be at least n ~(l°gn) = 2 (2(l°g2n). See [ 2 - 6 ] . The problem o f constructing finite packings with a large minimum kissing number received much less attention. In this short note we consider this problem and construct finite packings in [R~ with much higher kissing numbers than those o f the known lattice packings. For each (proper) power of 4, n the kissing number o f our packing in R" exceeds 2 '/~. To do so, we construct, for each integer k>~2, a linear, binary, error correcting code o f length n = 4 k, dimension k(2 k l + 1) and minimum distance n/4 in which the number of words of minimum Hamming weight is
(2k)
(2 a - 1) 2~_1
I
> 2 2k = 2 ~/n.
E-mail: [email protected].
0012-365X/97/$17.00 Copyright @ 1997 Elsevier Science B.V. All rights reserved PII SO0 I 2-365X(97)0007 I-X
N. Alon/DiscreteMathematics175 (1997) 249-251
250
It is easy to check that the finite collection of balls o f radius x/~/4 centered at all the vectors o f this linear code (considered as real vectors) forms a finite packing o f balls whose minimum kissing number is (2 k - l )
2k_i
> 2 2k = 2 v~.
Our construction is a concatenation code which is a variant o f one o f the constructions in [1]. Here are the details. Let F = GF(2 k) be the finite field with 2 k elements, where each member of F is represented by a binary vector o f length k. Construct the generating matrix A o f a linear code over GF(2) as follows. A has k(2 k-1 + 1) rows and 4 k columns. Let vl . . . . . v~ be a basis o f GF(2 ~) over GF(2). The rows o f A are indexed by pairs (i,j) with 0~
k
i=0 j=l
Note that the sum 2k-I
k
Z Z Ei,jl)jXi i=0 j= 1
is simply a polynomial of degree at most 2 k-1 over F, and therefore it is zero for at most 2 k-1 different values of x in F. If it is not zero for some x then as y varies exactly half o f the elements y give a 0 inner product with it and exactly half give a 1 product. Thus the Hamming weight is determined by the number o f zeros (in F ) o f the polynomial. There are only 2 k-I + 1 distinct weights and the minimum weight is obtained from polynomials with 2 k-l roots. The number o f such polynomials is precisely (2 k - l )
2k_t
'
where the 2 k - 1 factor comes from the fact that the polynomials are not necessarily monic, and the binomial coefficient is the number o f ways to choose the roots o f the polynomial. This proves the desired result and hence supplies the desired packings as well. Note that by letting y in the construction above vary over F - {0} instead of F we get a related code o f length n = 4 k - 2 k and the same dimension and weight distribution as the above one. Note also that by changing the upper bound for i in the definition o f the generating matrix A from 2 k-I to an arbitrary parameter h ( < 2 k) we get other codes with interesting parameters. A similar construction works, of course, for fields of other characteristics.
N. Alon/Discrete Mathematics 175 ( 1 9 9 7 ) 2 4 9 ~ 5 1
251
It would be interesting to decide if there is a linear binary code of length n in which the number of words of minimum Hamming weight is 2 al"). It would also be interesting to decide if there is a lattice packing in R" whose kissing number is 2 Q°'). We strongly suspect that such codes and such lattice packings do exist.
Acknowledgements I would like to thank N. Sloane for pointing out an error in a previous version of this note and for helpful comments.
References [1] N. Alon, O. Goldreich, J. Hastad, R. Peralta, Simple constructions of almost k-wise independent random variables, in: Proc. 31st IEEE FOCS, St. Louis, Missouri, IEEE, 1990, pp. 544-553; also in: Random Structures and Algorithms 3 (1992) 289-304. [2] E.S. Barnes, N.J.A. Sloane, New lattice packings of spheres, Canad. J. Math. 35 (1983) 30-41. [3] E.S. Barnes, G.E. Wall, Some extreme forms defined in terms of Abelian groups, J. Austral. Math. Soc. I (1959) 47~53. [4] A. Bos, J.H. Conway, N.J.A. Sloane, Further lattice packings in high dimensions, Mathematika 29 (1982) 171-180. [5] J.H. Conway, N.J.A. Sloane, Sphere Packings, Lattices and Groups, Springer, Berlin, 1988. [6] J. Leech, Some sphere packings in higher space, Canad. J. Math. 16 (1984) 657~582.