Edge codes constructed from unicyclic graphs
2026-08-31 • Information Theory
Information Theory
AI summaryⓘ
The authors build on Jaramillo-Velez's work, who created edge codes from the edges of graphs. While Jaramillo-Velez studied these codes for trees, this paper focuses on codes from unicyclic graphs, which have exactly one cycle. They found that understanding these codes is tricky when the cycle is even-length because the code’s properties depend on both the cycle's length and the size of the underlying field used in the construction. This reveals new complexities not seen in the tree case.
edge codestoric codesevaluation codesunicyclic graphshypergraphminimum distanceweight distributionbase fieldcycle lengthgraph theory
Authors
Sara Asensio, Giulia Gaggero, Naveena Ragunathan, Abhilash Saha, Adam Van Tuyl
Abstract
Jaramillo-Velez recently introduced edge codes, a new class of toric evaluation codes constructed from the edges of a (hyper)graph $\mathcal{H}$. In the case that $\mathcal{H}$ is a tree, Jaramillo-Velez computed both the minimum distance and the weight distribution of the associated code. In this paper, we study edge codes associated to unicyclic graphs. Our most striking result is that computing the parameters of these codes is subtle in the case that the induced cycle has an even length because these values will depend on certain conditions regarding the length of the cycle and the size of the base field.