A $q$-Polymatroid Framework for Information Leakage in Secure Linear Network Coding
Information Theory
2026-01-13 v1 math.IT
Abstract
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 -polymatroid associated with the underlying rank-metric code pair. Building on this connection, we introduce the notions of -polymatroid ports and -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 -analogue of the Brickell--Davenport theorem.
Keywords
Cite
@article{arxiv.2601.07567,
title = {A $q$-Polymatroid Framework for Information Leakage in Secure Linear Network Coding},
author = {Eimear Byrne and Johan Vester Dinesen and Ragnar Freij-Hollanti and Camilla Hollanti},
journal= {arXiv preprint arXiv:2601.07567},
year = {2026}
}