In this talk, we describe our recent efforts to construct a deterministic sequence of Low Density Parity Check codes that approaches the capacity of Binary Erasure Channels in the block-error rate sense. The main idea is to use large-girth constructions with optimized protographs. Protographs allow the use of degree-2 nodes (important for approaching capacity), while retaining a double-exponential fall in message erasure probability with iterations. In large-girth constructions, this guarantees an exponential fall in block-erasure probability with blocklength. By optimization, we are able to obtain small protographs with thresholds close to capacity under exact density evolution (threshold 0.493 for rate-1/2). The same large-girth construction is applicable to spatially coupled ensembles.