Lattices, linear codes, and invariants. I (Q2756714)

This is the first of a two-part article dealing with lattice packings and linear error-correcting codes. In this part, the author surveys known results concerning the sphere packing problem, including the connection between sphere packings and lattices, as well as their connection to theta functions and modular forms. NEWLINENEWLINENEWLINEAlthough the article is rather compact, the results are interesting and well developed. For example, the author explains how to identify the homothety classes \(\Lambda_n\) of all lattices of full rank in \(\mathbb{R}^n\) with the double coset space \(\text{SO}_n(\mathbb{R})\backslash \text{SL}_n(\mathbb{R})/ \text{GL}_n(\mathbb{Z})\). This leads to a natural choice of measure on \(\Lambda_n\) induced by invariant Haar measures on \(\text{SL}_n(\mathbb{R})\) and \(\text{SO}_n(\mathbb{R})\). As it is known, \(\Lambda_n\) has finite measure for all \(n\). Thus integration leads to a well-defined notion of averaging over \(\Lambda_n\), so that it makes sense to speak of a ``random'' lattice. After a short discussion on bounds for lattices, the author concludes that we cannot pack \(n\)-dimensional spheres significantly better than is achieved by a random lattice. NEWLINENEWLINENEWLINEClearly, this article serves as a rich introduction to the theory of sphere packings and related topics. The interested reader can obtain further information in e.g. [\textit{J. H. Conway} and \textit{N. J. A. Sloane}, Sphere packings, lattices and groups (Springer, New York) (1999; Zbl 0915.52003)].