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