High-dimensional (HD) chaotic map has wide applications in various research fields such as neural networks and secure communication. Designing HD chaotic maps with expected dynamics and robust hyperchaotic behaviors is an interesting but challenging topic. In this article, we propose an $n$ -dimensional hyperchaotic map $(n$ D-HCM) generation method on the basis of the Gershgorin-type theorem. First, the general form of the proposed $n$ D-HCM is built using $n$ parametric polynomials. Then, the entity and coefficient parameter matrices are configured according to the Gershorin-type theorem. Theoretical analysis shows that the generated $n$ D-HCM has $n$ positive Lyapunov exponents and thus can show robust hyperchaotic behaviors. Two examples of hyperchaotic map with specified equations are provided and their properties are analyzed to show the availability of the proposed method. Performance evaluations display that our $n$ D-HCM possesses abundant properties and complex behaviors, and it can outperform some representative HD chaotic maps. Moreover, to show the application of our $n$ D-HCM, we apply it to a secure communication scheme and the experimental results exhibit that it shows much better performance than these representative HD chaotic maps in resisting transmission noise.