Applications of Coding Theory to the Construction of Modular Lattices

✍ Scribed by Christine Bachoc

• Publisher: Elsevier Science
• Year: 1997
• Tongue: English
• Weight: 439 KB
• Volume: 78
• Category: Article
• ISSN: 0097-3165
• DOI: 10.1006/jcta.1996.2763

✦ Synopsis

We study self-dual codes over certain finite rings which are quotients of quadratic imaginary fields or of totally definite quaternion fields over Q. A natural weight taking two different nonzero values is defined over these rings; using invariant theory, we give a basis for the space of invariants to which belongs the three variables weight enumerator of a self-dual code. A general bound for the weight of such codes is derived. We construct a number of extremal self-dual codes, which are the codes reaching this bound, and derive some extremal lattices of level l=2, 3, 7 and minimum 4, 6, 8.

1997 Academic Press

1. Introduction

Most of the lattices known for their good density share the following property: they are l-modular for a certain level l equal to 1 or a prime number. This means, following [Q4], that they are even lattices such that a similarity of ratel sends their dual lattice to themselves. This definition includes the even unimodular lattices, and also famous lattices like the Coxeter Todd lattice of dimension 12 and level 3 and the Barnes Wall lattices which are, after rescaling, alternatively 2-modular or unimodular.

Such lattices appear naturally in the following situation: let K be either a number field with complex multiplication, or a quaternion field defined over a totally real number field with all its infinite places ramified in K. We denote by x Ä xÄ the canonical conjugation on K. Let V be a (left) K-vector space of finite dimension, endowed with a non degenerate hermitian form article no. TA962763 92 0097-3165Â97 25.00