Matrix Groups Related to the Quaternion Group and Spherical Orbit Codes

  • Authors:

  • A. Ya. Dorofeev;L. S. Kazarin;V. M. Sidelnikov;M. E. Tuzhilin

  • Affiliations:

  • Department of Math & Mechanics, Chair of Discrete of Mathematics, Moscow State University, Moscow, Russia 119899;Department of Math & Mechanics, Chair of Discrete of Mathematics, Moscow State University, Moscow, Russia 119899;Department of Math & Mechanics, Chair of Discrete of Mathematics, Moscow State University, Moscow, Russia 119899;-

  • Venue:
  • Designs, Codes and Cryptography
  • Year:
  • 2005

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Abstract

We consider a finite matrix group $${\cal N} \subset GL_8(\mathbb{R})$$ with 34· 216 elements, which is a subgroup of the infinite group $$\langle{\cal Q} , C\rangle \subset GL_8(\mathbb{C})$$ , where $${\cal Q}$$ is the regular representation of the quaternion group and C is a matrix that transforms the regular representation Q to its cellwise-diagonal form. There is a number of ways to define the matrix C. Our aim is to make the group $$\langle {\cal Q}, C\rangle$$ similar in a certain sense to a finite group. The eventual choice of an appropriate matrix C done heuristically.We study the structure of the group $${\cal N}$$ and use this group to construct spherical orbit codes on the unit Euclidean sphere in R8. These codes have code distance less than 1. One of them has 32· 28 = 2304 elements and its squared Euclidean code distance is 0.293.