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Spherical codes with prescribed signed permutation automorphisms inside shells of low-dimensional integer lattices

Ganzhinov Mikhail, Östergård Patric R. J

Abstract

Let $\textrm{S}(n,t,k)$ be the maximum size of a code containing only vectors of the $k$th shell of the integer lattice $\mathbb{Z}^n$ such that the inner product between distinct vectors does not exceed $t$. In this paper we compute lower bounds for $\textrm{S}(n,t,k)$ 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.

Spherical codes with prescribed signed permutation automorphisms inside shells of low-dimensional integer lattices

Abstract

Let be the maximum size of a code containing only vectors of the th shell of the integer lattice such that the inner product between distinct vectors does not exceed . In this paper we compute lower bounds for for small values of , and 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.
Paper Structure (4 sections, 2 theorems, 4 equations, 12 tables)

This paper contains 4 sections, 2 theorems, 4 equations, 12 tables.

Table of Contents

  1. Introduction
  2. Integer lattices
  3. Code construction
  4. Results

Key Result

Proposition 1

Linear maps defined by integer matrices with orthogonal rows of squared norm $k\in\mathbb{N}$ embed the shell $s_m$ into the shell $s_{km}$ for all $m\in\mathbb{N}$. This embedding preserves angles between vectors.

Theorems & Definitions (3)

  • Proposition 1
  • Proposition 2
  • proof