Mathematics Colloquia and Seminars

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Description

A number of particularly interesting low-dimensional codes and lattices have the extra property of being equal to (or, for lattices, similar to) their duals; as a result, it is natural to wonder to what extent self-duality constrains the minimum distance of such a code or lattice. The first significant result in this direction was that of Mallows and Sloane, who showed that a /doubly-even/ self-dual binary code of length n has minimum distance at most 4[n/24]+4, and with Odlyzko, obtained an analogous result for lattices. Without the extra evenness assumption, they obtained a much weaker bound; in fact, as I will show, this gap between singly-even and doubly-even codes is illusory: the bound 4[n/24]+4 holds for essentially all self-dual binary codes. For asymptotic bounds, the best result for doubly-even binary codes is that of Krasikov and Litsyn, who showed d<=D n+o(n)$ where D=(1-5^(-1/4))/2~0.165629. I'll discuss a different proof of their bound, applicable to other types of codes and lattices, in particular showing that for any positive constant c, there are only finitely many self-dual binary codes satisfying d>=D n-c n^(1/2).

3:45 Refreshments will be served before the talk in 551 Kerr Hall