arXiv subjects
Publications and source records attributed to Johan Vester Dinesen.
The connection between secret sharing and matroid theory is well established. In this paper, we generalize the concepts of secret sharing and matroid ports to $q$-polymatroids. Specifically, we introduce the notion of an access structure on a vector space, and consider properties related to duality, minors, and the relationship to $q$-polymatroids. Finally, we show how rank-metric codes give rise to secret sharing schemes within this framework.
We consider the problem of determining the expected dimension of the star product of two uniformly random linear codes that are not necessarily of the same dimension. We use a correspondence between the star product and the evaluation of bilinear forms to provide an explicit lower bound on the expected star product dimension. We prove that the expected dimension asymptotically reaches its maximum possible value as the field size increases. Furthermore, we show that the same maximal dimension is achieved asymptotically as the code dimensions increase, subject to a condition bounding their relative growth rates. We also analyze the variance of the star product dimension, providing explicit asymptotic upper bounds. Finally, we discuss the implications of these results for private information retrieval, secure distributed matrix multiplication, quantum error correction, and cryptanalysis.
We define almost affine vector rank-metric codes as subsets $\mathcal{C}\subseteq \mathbb{F}_{q^m}^n$ whose canonical projections have cardinalities that are powers of $q^m$, and prove that they naturally induce $q$-matroids. We establish that the operations of puncturing and shortening correspond to restriction and contraction of the $q$-matroid, and show that the rank-weight and formal dual distance distributions are determined by the induced $q$-matroid. We briefly discuss applications to perfect $q$-matroid ports in linear network coding, and show that disconnected $q$-matroids need not induce disconnected ports. Finally, we show that certain Additive Generalized Twisted Gabidulin codes yield direct examples of strictly almost affine rank-metric codes, alongside a separate construction derived from proper finite semifields.
We study information leakage in secure linear network coding schemes based on nested rank-metric codes. We show that the amount of information leaked to an adversary that observes a subset of network links is characterized by the conditional rank function of a representable $q$-polymatroid associated with the underlying rank-metric code pair. Building on this connection, we introduce the notions of $q$-polymatroid ports and $q$-access structures and describe their structural properties. Moreover, we extend Massey's correspondence between minimal codewords and minimal access sets to the rank-metric setting and prove a $q$-analogue of the Brickell--Davenport theorem.