eSeminar: On the code equivalence problem in rank metric

Vortrag von Dr. Alain Couvreur

Datum: 20.05.20  Zeit: 16.00 - 17.00  Raum:

(**This eSeminar will take place on Zoom, using same meeting details as previous seminars. If you do not have meeting details, please contact karan.khathuria@math.uzh.ch **)
The code equivalence problem can roughly be stated as follows : "Given two codes \(C_1\), \(C_2\), is there an isometry \(\phi\) of the ambient space such that \(\phi(C_1) = C_2\)?" In Hamming metric, this problem has been intensively studied in the last decades, with in particular the {\it support splitting algorithm} by N. Sendrier which solves this problem in the generic case when the isometry \(\phi\) is a permutation. On the rank metric side, the linear isometries of the ambient space are classified and various algebraic invariants related to a given matrix space have been identified. In this talk, we will focus on the algorithmic aspects of the code equivalence problem in rank metric by focusing on three versions: 1. \(\mathbb{F}_{q^m}\)--linear codes with a vector representation 2. \(\mathbb{F}_{q^m}\)--linear codes with a matrix representation 3. Non structured matrix spaces. We propose efficient algorithms to solve versions (1) and (2) of the problem. Then we prove that (3) is at least as hard as the monomial equivalence problem in Hamming metric. This is a work in progress in collaboration with Thomas Debris-Alazard (Royal Holloway, London) and Philippe Gaborit (University of Limoges).