Lattices, Linear Codes, and Invariants, Part II, Volume 47, Number 11

1382 NOTICES OF THE AMS VOLUME 47, NUMBER 11 I n Part I of this article we introduced “lattices” L ⊂ Rn and described some of their relations with other branches of mathematics, focusing on “theta functions”. Those ideas have a natural combinatorial analogue in the theory of “linear codes”. This theory, though much more recent than the study of lattices, is more accessible, and its numerous applications include the construction and analysis of many important lattices. A lattice L is a special kind of subgroup of the additive group Rn; in Part I we associated to L a theta function, which is a generating function for the lengths of the vectors of L. A linear code C , to be defined below, is a subgroup of a finite additive group Fn. Analogous to the theta functions of lattices are “weight enumerators” of codes, which are generating functions for the coordinates of elements of C . We shall see how most of the uses and properties of theta functions that we described in Part I have counterparts in the setting of weight enumerators. In particular, just as the theta function θL(τ) associated to a “self-dual lattice” L is invariant under certain fractional linear transformations of the variable τ, we shall encounter “self-dual codes” C with weight enumerators invariant under certain linear transformations of the variables. This invariance yields much information about C , and about an associated self-dual lattice LC and its theta function. As was true of Part I, there is little if any of the mathematics described herein for whose discovery I can claim credit. The one possible exception is the use of theta functions near the end to relate the occurrences of the groups SL2(Z/pZ) and SL2(Z) in the functional equations for weight enumerators and lattices respectively; I know of no published statement of this observation, though it may well be common knowledge in some circles. All other results and ideas I have attributed when their source is known to me, and I apologize in advance for any misor missing attribution.