This study presents a novel algorithm for transforming binary linear codes with parameters (n,k,d) into a bipartite graph representation. The proposed method explicitly represents each codeword as a node, enabling a complete structural visualization of the code. The algorithm is implemented and its computational performance is analyzed, demonstrating linear complexity with respect to code length (n) and exponential complexity with respect to dimension (k). The correctness and interpretability of the approach are illustrated using representative examples of (4,2) and (6,3) linear codes, where the resulting graphical structures reveal clear and meaningful patterns. In addition, the proposed representation is shown to be information-complete and to preserve code equivalence through graph isomorphism. The representation is particularly well-suited for integration with modern graph-based machine learning techniques, such as Graph Neural Networks, where structural information plays a central role in learning. Furthermore, the scalability characteristics of the algorithm make it applicable to a wide range of code parameters, while maintaining consistency in representation. To further assess the effectiveness of the proposed algorithm, it is compared with existing methodologies, including Trellis and Tanner graph representations, demonstrating advantages in structural analysis, effectiveness for graph-based learning, and its unique representation of the zero codeword. This framework therefore serves as a foundation for structural analysis of linear codes, facilitates equivalence testing, and is naturally suited for integration with graph-based machine learning models.
Olaewe et al. (Tue,) studied this question.