  Generalized product codes (GPCs) are extensions of product codes (PCs) where
code symbols are protected by two component codes but not necessarily arranged
in a rectangular array. We consider a deterministic construction of GPCs (as
opposed to randomized code ensembles) and analyze the asymptotic performance
over the binary erasure channel under iterative decoding. Our code construction
encompasses several classes of GPCs previously proposed in the literature, such
as irregular PCs, block-wise braided codes, and staircase codes. It is assumed
that the component codes can correct a fixed number of erasures and that the
length of each component code tends to infinity. We show that this setup is
equivalent to studying the behavior of a peeling algorithm applied to a sparse
inhomogeneous random graph. Using a convergence result for these graphs, we
derive the density evolution equations that characterize the asymptotic
decoding performance. As an application, we discuss the design of irregular
GPCs employing a mixture of component codes with different erasure-correcting
capabilities.
