Packings with large minimum kissing numbers

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 √ n others. A packing of balls in the Euclidean space is a finite or infinite collection of pairwise disjoint open unit balls in Rn. It is called a lattice packing if the centers of the balls form a lattice in Rn. The minimum kissing number of a packing is the minimum number of balls touching a given one. Note that for a lattice packing this is simply the number of balls touching any given one, since every ball touches the same number of others. The problem of existence and construction of lattice packings with high kissing numbers received a considerable amount of attention, and there are several known constructions that show that the kissing number of a lattice packing of balls in Rn may be at least nΩ(logn) = 2Ω(log 2 n). See [3], [4], [2], [6], and [5]. The problem of 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 Rn with much higher kissing numbers than those of the known lattice packings. For each (proper) power of 4, n the kissing number of our packing in Rn exceeds 2 √ n. To do so, we construct, for each integer k ≥ 2, a linear, binary, error correcting code of length n = 4k, dimension k(2k−1 + 1) and minimum distance n/4 in which the number of words of minimum Hamming weight is (2 − 1) ( 2k 2k−1 ) > 2 k = 2 √ . It is easy to check that the finite collection of balls of radius √ n/4 centered at all the vectors of this linear code (considered as real vectors) forms a finite packing of balls whose minimum kissing number is (2 − 1) ( 2k 2k−1 ) > 2 k = 2 √ . Our construction is a concatenation code which is a variant of one of the constructions in [1]. Here are the details. Let F = GF (2k) be the finite field with 2k elements, where each member of F is represented by a binary vector of length k. Construct the generating matrix A of a linear code over GF (2) as follows. A has k(2k−1 + 1) rows and 4k columns. Let v1, . . . , vk be a basis of ∗Department of Mathematics, Raymond and Beverly Sackler Faculty of Exact Sciences, Tel Aviv University, Tel Aviv, Israel. Email: [email protected].