Séminaire Théorie des Nombres

In this talk, we investigate intersecting codes. In the Hamming metric, these are codes where two nonzero codewords always share a coordinate in which they are both nonzero. Based on a new geometric interpretation of intersecting codes, we are able to provide some new lower and upper bounds on the minimum length $i(k, q)$ of intersecting codes of dimension k over $\mathbb{F}_q$, together with some explicit constructions of asymptotically good intersecting codes. We relate the theory of intersecting codes over $\mathbb{F}_q$ with the theory of $2$-wise weighted Davenport constants of certain groups, and to nonunique factorization theory. Finally, we will present intersecting codes in the rank metric.

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During this talk I will present a work in progress, joint with Félix Baril-Boudreau and Alexandre Benoist on the conjecture by Lang and Trotter that generalizes to elliptic curves Artin's conjecture on primitive roots.