Spherical Codes With Prescribed Signed Permutation Automorphisms Inside Shells of Low-Dimensional Integer Lattices

Let <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\textrm {S}(n,t,k)$ </tex-math></inline-formula> be the maximum size of a code containing only vectors of the kth shell of the integer lattice <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\mathbb {Z}^{n}$ </tex-math></inline-formula> such that the inner product between distinct vectors does not exceed t. In this paper we compute lower bounds for <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\textrm {S}(n,t,k)$ </tex-math></inline-formula> for small values of n, t and k by carrying out computer searches for codes with prescribed automorphisms. We prescribe groups of signed permutation automorphisms acting transitively on the pairs of coordinates and coordinate values as well as other closely related groups of automorphisms. Several of the constructed codes lead to improved lower bounds for spherical codes.