Lattices, Linear Codes, and Invariants, Part I, Volume 47, Number 10

1238 NOTICES OF THE AMS VOLUME 47, NUMBER 10 H ow should 24-dimensional toy merchants most efficiently store their marbles? This is one rather fanciful statement of the “sphere packing problem” in R24. This problem is not just a plaything of high-dimensional Euclidean geometry: it relates to a surprising range of mathematical disciplines, pure as well as applied, including number theory, finite groups, orthogonal polynomials, and signal transmission. The same is true of the closely related discrete problem of error-correcting codes. This is already true for the important special cases of “lattice” packings and “linear codes”. The present article is a two-part series devoted to lattices, linear codes, and their relations with other branches of mathematics. Even a two-part series does not afford enough space to indicate all the mathematical disciplines relevant to the study of lattices and codes; we have chosen to focus our attention on certain invariants attached to lattices and codes. In each case these are invariant in two senses: they can (but do not always) distinguish nonisomorphic lattices or codes, and they can be written as generating functions that are invariant, or at least transform predictably, under certain transformations of the variables. Part I, in this issue, mainly concerns lattices, whose relevant invariants are “theta functions”. Linear codes, and their close connections with lattices, will be the theme of Part II. As is usually the case in expository works, very little in Part I (in fact nothing outside the parenthetical remark on [EOR]) is my own work. I have attributed all results and ideas whose authors are known to me and apologize in advance if I have misattributed anything, or did not give a source for a result whose origin I do not know or wrongly believed to be classical or well known. At any rate, I do not claim any such result as my own.