We construct six K3 sigma models from self-dual classical codes via the Dymarsky-Shapere lattice CFT framework, classify their local complementation orbits, and prove that D₄ triality endows the common discriminant group with F₄ field structure yielding the hexacode. We show that symmetry surfing across Kummer and Gepner models generates the full Mathieu group M₂₄ from a single 6A-class element, identifying the sextet obstruction as the precise barrier overcome by non-geometric models.
Sai Tilak Pally (Mon,) studied this question.