Additive codes arising from hypergraphs

Source abstract

We study the critical exponent of additive codes through an integer polymatroid associated with the code. We give a coding-theoretic proof of Whittle's Critical Theorem in this setting, a geometric description of the critical exponent in terms of $h$-projective systems, and general bounds, including an analogue of Kung's girth bound. We then study additive codes whose polymatroid is the hypergraphic polymatroid of a hypergraph $H$. For these codes the critical exponent turns to be determined by the weak chromatic number of $H$. If the code is faithful, then the minimum folded Hamming weight of the dual code is equal to the Berge girth of $H$. If $H$ is connected, the minimum distance is equal to the edge-connectivity of $H$. As a consequence, for $h\geq2$ we determine all such codes with connected $H$ that attain the Singleton bound, that is, all faithful hypergraphic additive quasi-MDS codes. We specialise the Griesmer and linear programming bounds to hypergraphic codes, we derive a lower bound on the minimum distance from the Laplacian eigenvalues of the weighted $2$-section of $H$, and we compare all these bounds computationally.