Connection to codes has emerged recently as a new tool to construct and study Narain CFTs with special properties. In the talk I will review this connection and argue that optimal theories, i.e. those maximizing the value of spectral gap for the given central charge, are code CFTs - meaning they can be constructed using codes. This applies to known optimal theories with c<=8 as well as to asymptotically large c. I will also discuss spinoff results, in particular construction of fake torus partition function Z(\tau, \bar \tau), which satisfies all properties of the 2d CFT torus partition function (modular invariance, discreteness and positive-definiteness of spectrum), yet can be shown not to be a partition function of any 2d theory.
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Slava Rychkov