We observe that every self-dual ternary code determines a holomorphic $\mathcal N=1$ superconformal field theory. This provides ternary constructions of some well-known holomorphic $\mathcal N=1$ superconformal field theories (SCFTs), including Duncan’s “supermoonshine” model and the fermionic “beauty and the beast” model of Dixon, Ginsparg, and Harvey. Along the way, we clarify some issues related to orbifolds of fermionic holomorphic CFTs. We give a simple coding-theoretic description of the supersymmetric index and conjecture that for every self-dual ternary code this index is divisible by $24$ ; we are able to prove this conjecture except in the case when the code has length $12$ mod $24$ . Lastly, we discuss a conjecture of Stolz and Teichner relating $\mathcal N=1$ SCFTs with Topological Modular Forms. This conjecture implies constraints on the supersymmetric indexes of arbitrary holomorphic SCFTs, and suggests (but does not require) that there should be, for each k, a holomorphic $\mathcal N=1$ SCFT of central charge $12k$ and index $24/\gcd (k,24)$ . We give ternary code constructions of SCFTs realizing this suggestion for $k\leq 5$ .

我们观察到每个自对偶三元码都决定了一个全纯 $\mathcal N=1$ 超共形场论。这提供了一些著名的全纯 $\mathcal N=1$ 超共形场论 (SCFT) 的三元构造，包括 Duncan 的“supermoonshine”模型和 Dixon、Ginsparg 和 Harvey 的费米子“美女与野兽”模型。在此过程中，我们澄清了与费米子全纯 CFT 的圆环相关的一些问题。我们给出了超对称索引的简单编码理论描述，并推测对于每个自对偶三元码，该索引可以被 $24$ 整除；我们能够证明这个猜想，除非代码长度 为 $12$ mod $24$ . 最后，我们讨论了 Stolz 和 Teichner 将 $\mathcal N=1$ SCFT 与拓扑模形式相关的猜想。这个猜想暗示了对任意全纯 SCFT 的超对称索引的约束，并建议（但不要求）对于每个k ，应该有一个中心电荷 $12k$ 和索引 $24/的全纯 $\mathcal N=1$ SCFT \gcd (k,24)$ 。 我们给出了实现$k\leq 5$ 建议的 SCFT 的三元代码结构。