With Volker Puppe I have shown that all codes come from 3-manifolds with involution. But explicit constructions were only known in special cases. There is a construction associating to an element in the mapping class group a 3-manifold with involution and Johannes Holke has proved that many codes of length up to 24 arise this way. To study cases where codes of length >24 occur it is very useful to have a simple criterion when the code is doubly even. This can be related to the existence of a Spin structure of a certain 4-manifold, which generalizes the Kummer surface. These manifolds are of separate interest. I will report about these recent results.

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