Characterize paving matroids with extended-subadditive generalized Hamming weights

Determine precisely which paving matroids M (matroids in which all circuits have size rk(M) or rk(M)+1) have generalized Hamming weight sequences {d_r(M)}_{r=1}^{|E|-rk(M)} that are extended subadditive; that is, satisfy both d_{i+j}(M) ≤ d_i(M)+d_j(M) whenever 1 ≤ i,j and i+j ≤ |E|−rk(M), and d_r(M)+d_{|E|-rk(M)}(M) ≤ d_t(M)+d_{|E|-rk(M)+r−t}(M) for every 1 ≤ r < t ≤ |E|−rk(M).

Background

The paper proves that for any paving matroid M of rank at least two, the generalized Hamming weights {d_r(M)} form a subadditive sequence and that this implies α(I_Δ(M)^{(r)})=d_r(M) for all r. It further shows that sparse paving matroids yield an extended subadditive sequence and thus explicit formulas for α(I_Δ(M)^{(s)}) and the Waldschmidt constant.

However, an example demonstrates that not every paving matroid has an extended subadditive sequence. This motivates a structural characterization of those paving matroids for which the stronger extended subadditivity holds.