Trygve Johnsen

Sudhir R. Ghorpade, Trygve Johnsen, Rati Ludhani et al.

We determine all the Betti numbers of the $q$-ary second order Reed-Muller codes of length $q^2$, and also of the elongations of matroids associated to these codes. We then use it to determine the higher weight spectra of these codes. As a special case, we recover some results of Kaplan and Matei about counting certain curves over finite fields with prescribed rational intersection points. In geometric terms, our results relate to the affine Veronesean by which we mean the image of the affine plane $\mathbb{A}^2$ under the quadratic Veronese embedding of $\mathbb{P}^2$ in $\mathbb{P}^5$. Indeed, finding the higher weight spectra, of the Reed-Muller code considered here, corresponds to determining the number of $\mathbb{F}_q$-rational points on all possible sections of this affine Veronesean by linear subvarieties of $\mathbb{P}^5$.

Duy Ho, Trygve Johnsen

The intersection of a linear code with its dual is called the hull of the code. It is known that, for classical linear codes under the Hamming-metric, the dimension of the hull can be reduced up to equivalence. This phenomenon leads to the so-called hull-variation problem formulated by Hao Chen in 2023. In this paper, we consider the analogous problem for vector rank-metric codes, along with their associated matrix codes and extended block codes. Our results include the fact that every vector rank-metric code over any finite field $\mathbb{F}_q$, in particular when $q=2$ or $q=3$, is equivalent to an LCD code.