Nonbinary code-based graphs and applications

This paper considers the notion of nonbinary (linear) codes and the notion of hypergraphs and makes a relation between two mathematical tools for real-world applications. We consider a nonbinary (linear) code as an underline set and construct a type of hypergraphs called [Formula: see text]-hypergraph ([Formula: see text]-hypergraph) and establish any given hypergraph is isomorphic to an [Formula: see text]-hypergraph ([Formula: see text]-hypergraph). This study introduces an equivalence relation on any given code as a positive equivalence relation based on the weight of codewords, generates quotient [Formula: see text]-hypergraph, and computes the cardinal of its hyperedges and Hamming distance of any given two code words of quotient [Formula: see text]-hypergraphs. Also, we introduce a binary relation on quotient [Formula: see text]-hypergraphs, convert them to simple graphs as code-based graphs, and prove that all graphs are positive code-based graphs. We characterize the minimum distance and maximum distance of codewords in hyperedges of code-based (hyper)graphs concerning linear codes. Finally, we design real problems (such as wireless sensor (hyper)networks and American standard code for information interchange) as complex hypernetworks (such that are based on the code words) as a quotient [Formula: see text]-hypergraph ([Formula: see text]-hypergraph) and convert them to the code-based graphs (such as wireless sensor networks).