To Generalized Weights... and Beyond: New (and Old) Invariants for Linear Codes
Talk by Andrea Di Giusto
Date: 02.04.25 Time: 15.15 - 16.00 Room: Y27H28
Generalized Hamming Weights (GHW) have seen a big rise in popularity since Victor Wei described their many properties in 1991, linking them to code performance on the wire-tap channel of type 2. Many equivalent definitions have been proposed, including one relating them to Optimal Linear Anticodes by Ravagnani (2016): Anticodes (codes whose dimension is equal to the maximal weight) can be used as a family of test codes to determine the GHW (when the base field is not the binary field). The properties of GHW can then be inferred by the properties of the family of Anticodes.In this talk, we further extend the approach to arbitrary families of test codes, focusing on a minimal set of assumptions yielding invariants with good duality properties (that is, similar to those proved by Wei for GHW). In doing so, we show that our approach is independent of the chosen metric: in particular, we recover in a unique result the duality of generalised weights in the Hamming and rank metrics. This level of generality also allows us to tackle the problem of duality of generalised weights in the sum-rank metric, by showing a first example of codes with nontrivial Hamming and rank metric parts for which the duality of generalised weights holds. Finally, we investigate the invariants obtained by using the family of Singleton-optimal codes (MDS/MRD codes) in place of Anticodes, highlighting similarities and differences between the two families that reflect on the properties of the obtained invariants. This is joint work with Elisa Gorla and Alberto Ravagnani.