Additive Combinatorics, Field Extensions, and Coding Theory.

8 June 2016
15:00
Gilles Zémor
Abstract
Additive combinatorics enable one to characterise subsets S of elements in a group such that S+S has
small cardinality. In particular a theorem of Vosper says that subsets of integers modulo a prime p
with minimal sumsets can only be arithmetic progressions, apart from some degenerate cases. We are
interested in q-analogues of these results, namely characterising subspaces S in some algebras such
that the linear span of its square S^2 has small dimension.
Analogues of Vosper's theorem will imply that such spaces will have bases consisting of elements in
geometric progression.
We derive such analogues in extensions of finite fields, where bounds on codes in the space of
quadratic forms play a crucial role. We also obtain that under appropriately formulated conditions,
linear codes with small squares for the component-wise product can only be generalized Reed-Solomon
codes.

Based on joint works with Christine Bachoc and Oriol Serra, and with Diego Mirandola.
  • Cryptography Seminar