We introduce CODD, a system for solving combinatorial optimization problems using decision diagram technology. Problems are represented as state-based dynamic programming models using the CODD language specification. The model specification is used to automatically compile relaxed and restricted decision diagrams that are embedded inside a branch-and-bound search process. We introduce abstractions that allow us to generically implement the solver components while maintaining overall execution efficiency. We demonstrate the functionality of CODD on a variety of combinatorial optimization problems and compare its performance to other state-based solvers as well as integer programming and constraint programming solvers. CODD provides competitive results and can outperform the other solvers, sometimes by orders of magnitude.