On Zero-Error Capacity of Graphs with One Edge
Qi Cao, Qi Chen, Baoming Bai
TL;DR
This work addresses the zero-error capacity problem for channels with memory represented by graphs, focusing on graphs that contain a single edge. It introduces a constructive method to build codes for a graph $G(\bm{u},\bm{v})$, establishing a lower bound $C(G)\ge -\log x^*$ where $x^*$ solves $x^{\ell(\bm{u_v})}+x^{\ell(\bm{v_u})}=1$, and proves this bound is tight when a degree-one vertex has identical symbols. The method yields an asymptotically optimal quasi-2-code $\mathcal{B}_n=\{\bm{u_v},\bm{v_u}\}^*\cap \mathcal{X}^n$ for suitable cases, and, for the binary channel with two memories, determines $C(G)$ exactly for 10 symmetry classes and provides bounds for the remaining Case 11. Together, these results offer a general, scalable approach to lower-bounding zero-error capacity for one-edge graphs and advance the understanding of memory-channel capacities in structured graph models.
Abstract
In this paper, we study the zero-error capacity of channels with memory, which are represented by graphs. We provide a method to construct code for any graph with one edge, thereby determining a lower bound on its zero-error capacity. Moreover, this code can achieve zero-error capacity when the symbols in a vertex with degree one are the same. We further apply our method to the one-edge graphs representing the binary channels with two memories. There are 28 possible graphs, which can be organized into 11 categories based on their symmetries. The code constructed by our method is proved to achieve the zero-error capacity for all these graphs except for the two graphs in Case 11.