Lattices over Integers of Number Fields and Self-Dual Codes

Abstract

Up to now, we have only considered binary codes. In this chapter we want to present a generalization of the results of Chapter 2 to self-dual codes over 𝔽_pp, where p is an odd prime number. The results which we want to discuss are due to G. van der Geer and F. Hirzebruch [Hir87, pp. 759-798]. In §1.3 we associated an integral lattice in ℝⁿ to a binary code of length n. In §5.2 we shall generalize this construction by associating a lattice over the integers of a cyclotomic field to a code over 𝔽_pp. In this section we shall study lattices over integers of cyclotomic fields. For the background on algebraic number theory see also [Sam67] and [ST87].