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.