Symmetries of discrete structures

Though perhaps not true in high dimensions our experience in small dimension shows that beautiful objects do have symmetries. Symmetries are my guide to construct extremal codes and lattices. The notion of extremality stems from a very fruitful connection between codes and invariant theory of finite groups and its analogous relation between lattices and modular forms.

Invariant theory allows to upper bound the error correcting properties of codes in certain families. Codes achieving this upper bound are called extremal. In 1973 Neil Sloane published a short note asking whether there is an extremal code of length 72. Since then many mathematicians search for such a code, developing new tools to narrow down the structure of its symmetry group. We now know that, if such a code exists, then it has at most 5 symmetries.

The methods for studying this question involve explicit and constructive applications of well known classical theorems in algebra and group theory, like Burnside's orbit counting theorem and quadratic reciprocity, as well as basic representation theoretic methods and tools from the theory of quadratic forms.

Similar methods have been recently developed to study extremal lattices admitting certain symmetries. In my talk I will survey the results obtained on extremal codes and lattices and give some nice examples of the use of symmetries to reduce the search space.