Martino Borello

LAGA, Université Paris 8

Extremal self-dual codes and their lattice counterparts.

Self-dual codes are among the most rigid objects in coding theory. The MacWilliams identity forces their weight enumerators to be invariant under a finite matrix group. As a consequence, Gleason’s theorem shows that the weight enumerator of a doubly-even self-dual code is a polynomial in those of the extended Hamming code of length 8 and the Golay code of length 24.

In these talks we present this invariant-theoretic approach via Molien series and derive the resulting bounds on the minimum distance, which lead to the notion of extremal codes. We then discuss the extremal codes of length 8, 24 and 48: their uniqueness, their automorphism groups, and the designs they carry by the Assmus-Mattson theorem.

Each step has a counterpart for even unimodular lattices: theta series, Poisson summation and Hecke’s theorem play the roles of weight enumerators, the MacWilliams identity and Gleason’s theorem. This parallel breaks down in length 72. An extremal even unimodular lattice of dimension 72 was constructed by Nebe, whereas the existence of an extremal self-dual [72,36,16] code, a question raised by Sloane in 1973, is still open.

We present the main approach to this problem, the study of the possible automorphisms of such a code, including some of our results, and conclude with a selection of open questions.

Frederique Oggier

University of Birmingham

TBA.

TBA

Wessel van Woerden

PQShield

TBA

TBA