New asymptotic bounds on the size of multiple packings of the Euclidean sphere
A finite set (spherical code) \(K^n \subset \mathbb{S}^{n-1}\) is an \(L\)-multiple packing of radius \(t\) if \(\max_{x \in \mathbb{R}^n} |B(x,t) \cup K^n| \leq L\). Asymptotic estimations on the maximal possible size of an \(L\)-multiple packing are stated as relations between the code rate \(R\), defined by \(\ln |K^n| \sim Rn\) and \[ t_L= \lim\sup_{n \to \infty} \sup_{|K^n| \geq exp(Rn)} \frac{\sup t_L(K^n)}{n}, \] where \(t_L\) is the supremum of \(t\) for the code \(K^n\). In the paper under review, the authors obtain new upper bounds on \(t_L\). The proof is based on linear programming bounds on the distance distributions of large spherical codes followed by a recursive procedure for finding a small simplex of \(L+1\) points of \(K^n\).
scientific article; zbMATH DE number 5979922