An Analysis of Fundamental Parameters of Graph-Based Linear Error- Correcting Codes

Abstract

In this work, we develop the linear codes generated by the incidence matrices of certain finite
graphs over the finite field ????????. The main objective is to determine the fundamental parameters of
the associated codes, namely the length, dimension, and minimum distance, and to examine the
influence of graph structures on the resulting coding properties. Both bipartite and non-bipartite
graphs are considered in this work. For the bipartite case, the Hypercube graph ????????, Gray graph,
and Nauru graph are examined, and the corresponding linear codes are shown to have parameters
[????2 ????−1, 2???? − 1, 2]????, [81, 53, 3]????, and [36, 23, 3]????, respectively. Further, for the non-bipartite
graphs ????????(5,2) and the Wagner graph, the associated linear codes are determined to have
parameters [15, 10, 3]???? and [12, 8, 3]????, respectively.