In this talk I will present some recent and ongoing work with Ashish Kakkar and Brian McPeak, where we describe a very explicit construction of a class of two-dimensional conformal field theories, denoted code CFTs. In the chiral case, code CFTs are constructed by compactifying n free bosons on a lattice, which in turn is defined from a classical error-correcting code via Construction A by Leech and Sloane. We show that constraints from higher-genus modular invariance on code CFTs can be recast into simple linear relations on the the higher-weight enumerator polynomial, which is a natural object from the code perspective. With this machinery at hand, we show that higher-genus modular invariance greatly reduces the number of seemingly consistent code CFT partition functions that were found by demanding modular invariance at genus one only. I will also cover some upcoming work, on the relation between quantum error-correcting codes and non-chiral (Narain) CFTs, and on averaging over code CFTs.