Uniquely optimal codes of low complexity are symmetric
This study addresses the fundamental question of whether optimal codes in compact metric spaces necessarily exhibit symmetry, focusing on low-complexity uniquely optimal encodings. Method: Integrating tools from metric geometry, extremal combinatorics, and group action theory, we develop constructive coding design techniques, symmetry detection algorithms, and rigorous optimality proofs. Contribution/Results: We formulate and systematically verify the universal conjecture that every low-complexity uniquely optimal code admits a nontrivial symmetry. Through comprehensive case studies on canonical spaces—including the sphere and torus—we empirically and theoretically confirm that symmetry is a necessary condition for uniqueness and optimality under low complexity constraints. Our work establishes, for the first time, a deep structural connection between the geometry of the underlying metric space and the symmetry properties of its optimal codes. This yields novel principled guidance for efficient encoding design, advancing both theoretical understanding and practical construction of optimal codes in geometric settings.