Sparse Modular Forms, Lattices, and Codes

Motivated by sparseness conditions for holographic CFTs, we investigate sparseness of modular forms, lattices, and codes. For this we investigate the free energy of such objects as their weight, dimension or size goes to infinity. We construct families of modular forms that are sparse, such as the Eisenstein series $E_{2k}(\tau)$. We then investigate lattices that come from codes and introduce a sparseness condition for such lattices. We investigate the limit of lattices constructed from self-dual Reed-Muller codes and provide evidence that they are sparse in this sense.