Quantum error-correction code discovery has relied on algebraic constructions with predetermined structure or computational search lacking equilibrium-topology analysis. We introduce a game-theoretic framework recasting code optimization as strategic interactions between competing objectives, where Nash equilibria systematically generate codes with desired properties. We validate the framework by demonstrating it rediscovers the optimal [[15,7,3]] quantum Hamming code in 20% of independent runs [A. R. Calderbank and P. W. Shor, Phys. Rev. A 54, 1098 (1996); A. Steane, Proc. R. Soc. Lond., Ser. A 452, 2551 (1996)] from competing objectives without predetermined algebraic structure, with equilibrium analysis providing transparent mechanistic insights into why this topology emerges. Applied across seven objectives [distance maximization, hardware adaptation, rate-distance optimization, cluster-state generation, surfacelike topologies, connectivity enhancement, and maximization of the quantum Fisher information (which quantifies, via the Cramér-Rao bound, the metrological sensitivity of the encoded codespace)], the framework generates distinct code families through objective reconfiguration rather than algorithm redesign. Scalability to hardware-relevant sizes is demonstrated at $n=100$ qubits, discovering codes including [[100,50,4]] with distance-4 protection and 50% encoding rate, with tractable cubic per-iteration complexity enabling discovery in under 1 h. This work opens research avenues at the intersection of game theory and quantum information, providing systematic, interpretable frameworks for quantum system design.

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• Quantum circuits
• Quantum engineering
• Quantum error correction
• Quantum metrology