Self-dual codes, lattices, and invariant theory

Abstract

One of the most interesting fields in all of mathematics concerns the interaction between the fields of integral lattices, modular forms, invariant theory, and error-correcting codes. There are several excellent presentations in the literature of this subject, for example, Conway and Sloane (Sphere Packings, Lattices and Groups, 3rd edn. Springer, Berlin, 1999), Ebeling (Lattices and codes, 2nd edn. Vieweg, 2002), Elkies (Not. Am. Math. Soc. 47:1238–1245, 2000) Sloane (Proc. Symp. Pure Math., vol. 34, pp. 273–308. AMS, Providence, 1979), and Brualdi, Huffman, and Pless (Handbook of Coding Theory. Elsevier, New York, 1998). Therefore, this chapter will be brief and refer to these works for details. Topics treated in this chapter include (a) invariant theory and the relationship with self-dual codes, (b) lattices and connections with binary codes, and (c) optimal, divisible, and extremal codes. Some open questions which arise are: Which polynomials F(x,y) occur as the weight enumerators of linear codes? Does there exist a binary self-dual [72,36,16] code? These questions and others are discussed below.